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Linear Algebra- Chapter 3 20080926

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Chapter 3 Elementary Matrix Operations and Systems of Linear Equation

3-1 Elementary Matrix Operations and Elementary Matrices

In this section we define the elementary operations that are used throughout the chapter

In subsequent sections we use these operations to obtain simple computational methods for

determining the rank of a linear transformation and the solution of a system of linear

equations

何謂「基本矩陣運算（Elementary matrix operation）」「基本矩陣（Elementary

matrix）」以及基本矩陣運算可用於計算線性轉換的 Rank 與解線性方程組

There are two types of elementary matrix operations ndash row operations and column

operations

DEFINTION 31 Elementary operator

Let A be an mtimesn matrix Any one of the following three operations on the rows

[columns] of A is called an elementary row [column] operation

何謂「基本矩陣運算（Elementary matrix operation）」

令 A 為一 mtimesn 的矩陣以下這三個運算可以稱為基本列或行運算

(1) TYPE 1 interchanging any two rows [columns] of A

將 A 的任兩行或任兩列互換

(2) TYPE 2 multiplying any row [column] o A by a nonzero scalar

將非零常數乘至 A 中某一列

(3) TYPE 3 adding any scalar multiple of a row[column] of A to another

row[column]

將 A 的某一列乘上任一常數後加到另外一列


2

EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A 經過四種不同型態的運算helliphellip

TYPE 1 Interchanging the second row of A with the first row The result matrix is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

210443213112

B

TYPE 2 Multiplying the second column of A by 3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431324361

C

TYPE 3 Adding 4 times the third row of A to the first row

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

21043112

127217M

TYPE 3 Adding -4 times the third row of M to the first row of M A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

Notice that if a matrix Q can be obtained from matrix P by means of an elementary row

operation the P can be obtained from Q by an elementary row operation of the same type

注意若矩陣 Q 經過基本列運算後得到矩陣 P則矩陣 P 亦可由同一 TYPE 的

基本列運算得到矩陣 Q

DEFINITION 32 Elementary matrix

An ntimesn elementary matrix is a matrix obtained by performing an elementary operation

on In The elementart matrix is said to be of type 1 2 or 3 according to whether the

elementary operation performed on In is type 1 2 or 3 operation respectively

「基本矩陣」係指在 In 上實施某一基本運算後所得的矩陣分別在 In 上實施

TYPE 123 基本運算後所得基本矩陣稱為 TYPE 123


3

For example interchanging the first rows of I3 produces the elementary matrix

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100001010

E 為基本矩陣（Elementary matrix）係由 I3 的前二行互調或前兩列

互調取得

Note that E can be obtained by interchanging the first two columns of I3 In fact any

elementary matrix can be obtained in at least two ways ndash either by performing an elementary

row operation on In or by performing an elementary column operation on In Similarly

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus

100010201

也是基本矩陣（Elementary matrix）此矩陣可由兩種管道取得

is an elementary matrix since it can be obtained from I3 by an elementary column

operation of type 3 (adding -2 times the first column of I3 to the third column)

Theorem 31 Let AisinMmtimesn(F) and suppose that B is obtained from A by performing an elementary

row [column] operation Then there exists an mtimesm [ntimesn] elementary matrix E such that B

= EA [B = AE] IN fact E is obtained from Im [In] by performing the same elementary row

[column[ operation as that which was performed on A to obtain B Conversely if E is an

mtimesm [ntimesn] elementary matrix then EA [AE] is the matrix obtained from A by performing

the same elementary row [column] operation as that which produces E from Im [In]

令 AisinMmtimesn(F)並假設 B 是由 A 經實施某一基本列【或行】運算後所得到的矩

陣則存在一個 mtimesm 【或 ntimesn】基本矩陣 E使得 B = EA【或 B = AE】E 矩陣是

由 Im 【或 In】經實施與「由 A 得到 B」相同的基本列【或行】運算後所得到的矩

陣反之若 E 是一個 mtimesm【或 ntimesn】的基本矩陣則 EA【或 AE】是由 A 經過與

「由 Im 【或 In】得到 E」相同的基本列【或行】運算後所得到的矩陣

A Elementary row [column] operations B

Im Elementary row [column] operations E

B = EA [= AE]


4

EXAMPLE 2 Consider the matrices A and B in Example 1 In this case B is obtained from A by

interchanging the second row of A with the first row Performing this same operation on I3

we obtain the elementary matrix

範例 1A 經過基本列運算（Interchanging the second row of A with the first

row） BIm 經過基本列運算（Interchanging the second row of A with the first

row） E則 B 亦可由 B = EA 取得

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100001010

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus==

210443213112

EAB

Consider the matrices A and C in Example 1 In this case C is obtained from A by

multiplying the second column of A by 3 Performing this same operation on I4 we obtain

the elementary matrix

範例 1A 經過基本行運算（Multiplying the second column of A by 3） C

In 經過基本行運算（Multiplying the second column of A by 3） E則 C 亦可由 C

= AE 取得

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000010000300001

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

C = AE = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

210431324361

Theorem 32 Elementary matrices are invertible and the inverse of an elementary matrix is an

elementary matrix of the same type

基本矩陣為可逆矩陣且基本矩陣的反矩陣亦是同一型的基本矩陣

【Proof】

Let E be an elementary ntimesn matrix Then E can be obtained by an elementary row


5

[column] operation on In By reversing the steps used to transform In into E we can

transform E back into In The result is that In can be obtained from E by an elementary row

[column] operation of the same type By Theorem 31 there is an elementary matrix E

such that nIEE = Therefore E is invertible and 1EE minus=

令 E 是 ntimesn 的基本矩陣E 可由 In 經實施一基本列或行運算取得若把用來實

施 In E 的運算對 E 實施亦可得到 In依據 Theorem 31 可知存在一基本矩陣E

使得 nIEE = 因此 E 為可逆且 1EE minus=

Elementary Matrices Elementary Matrices 1414

An elementary matrix is one that can be obtained from the identity matrix In through a single elementary row operation

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

3I⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

010100001

1ER2 harr R3

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100050001

2E5R25R2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100012001

3ER2+ 2R1R2+ 2R1

Operation TOperation T


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

ihgfedcba

A

AEAfedihgcba

1

010100001

=sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡R2 harr R3

AEAihgfed

cba

2

100050001

555 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡5R2

AEAihg

cfbeadcba

3

100012001

222 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+++

R2+ 2R1


Each elementary matrix is invertible

1221EI

RR +asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010021

1E

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

I

IE 2R21R1 minus

asymprArr IEE =12 ie

22R21REI

minusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minus=

100010021

E 1


If A and B are row equivalent matrices and A is invertible then B is invertibleSince A and B are row equivalent there exits a sequence of row operations T1hellipTn such that B=TnT1ALet the elementary matrices of these operations be E1hellipEn Thus B=En hellip E2 E1 AThe matrix A En hellipE2 E1 are all invertible

( )( )

1

n

1

3

1

2

1

111

11

1n

1

n

1

3

1

2

1

1

1

n

1

3

1

2

1

11

EEEEABBAEE

EEEAEEEEEA

minusminusminusminusminusminus

minusminus

minusminusminusminusminusminusminusminus

=rArr==

=

3-2 The Rank of a Matrix and Matrix Inverse

In this section we define the rank of a matrix We then use elementary operations to

compute the rank of a matrix and a linear transformation The section concludes with a

procedure for computing the inverse of an invertible matrix

何謂「矩陣的秩（Rank）」利用基本行列運算來計算矩陣與線性轉換的秩本節

還包括介紹如何計算可逆矩陣的反矩陣


6

DEFINITION 33 Rank of a Matrix

If AisinMmtimesn(F) we define the rank of A denoted rank(A) to be the rank of the linear

transformation LA Fn Fm

若 AisinMmtimesn(F)則矩陣 A 的秩為線性轉換 LA Fn Fm的秩註記為 rank(A)

An ntimesn matrix is invertible if and only if its rank is n

一個 ntimesn 的矩陣為可逆的『若且唯若』條件為該矩陣的 Rank 為 n

Every matrix A is the matrix representation of the linear transformation LA with respect

to the appropriate standard ordered bases Thus the rank of the linear transformation LA is the

same as the rank of one of its matrix representation namely A

每一個矩陣 A 是線性轉換 LA 對於某個標準有序基底的矩陣表示式因此該線性轉

換 LA的 Rank 就等於矩陣表示式 A 的 Rank即 rank(LA) = rank(A)

DEFINITION 29

Let V and W be vector spaces and let T V W be linear If N(T) and R(T) are finite-

dimensional then we define the nullity of T denoted nullity(T) and the rank of T denoted

rank(T) to be the dimensions of N(T) and R(T) respectively

V 與 W 為向量空間且 T V W 為線性轉換

定義 T 的核次數（Nullity）nullity(T) = dimensions of N(T)

定義 T 的秩（Rank）rank(T) = dimension of R(T)

核次數（Nullity）與秩（Rank）分別代表 N(T)與 R(T)的維度

DEFINITION 211

Let β = u1 u2hellip un be an ordered basis for a finite-dimensional vectors space V

For xisinV let a1 a2hellip an be the unique scalars such that

令 β = u1 u2hellip un為有限向量空間 V 的有序基底則存在唯一的一組純量 a1

a2hellip an使得 V 內任意向量均可表達成該組基底的線性組合

sum=

=n

1iiiuax

we define the coordinate vector of x relative to β denoted β[x] by


7

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=β

n

2

1

a

aa

[x] 稱為 x 相對於有序基底 β的座標向量

DEFINITION 212

Suppose that V and W are finite-dimensional vector spaces with ordered bases β = v1

v2hellip vn and γ = w1 w2hellip wn respectively Let T V W be linear Then for each j 1

≦j≦n there exist unique scalars aijisinF 1≦i≦m such that

sum==

m

1iiijj wa)vT( for 1≦j≦n

We call the mtimesn matrix A defined by Aij = aij the matrix representation of T in the ordered bases β and γ and write γ

β= [T]A

If V = W and β = γ then we write β= [T]A

設 V 與 W 分別為有限維度的向量空間β = v1 v2hellip vn與 γ = w1 w2hellip

wn分別為 V 與 W 的有序基底且 T V W 為 V 映至 W 的線性轉換

對每一個 j（1≦ j≦n）而言存在唯一的純量 aij isinF（1≦ i≦m）使得

sum==

m

1iiijj wa)vT(

將 mtimesn 的矩陣 A 定義為 Aij = aij並稱呼 A 為線性轉換 T 的矩陣表達方式在

V 與 W 分別以 β 與 γ作為有序基底下A 可註記為 γβ= [T]A

若 V = W 且 β = γ則 β= [T]A

提示 sum==

m

1iiijj wa)vT( 為「將 vj的像 T(vj)表達成有序基底 γ的線性組合」

DEFINITION 217

Let A be an mtimesn matrix with entries from a field F We denote by LA the mapping LA

Fn Fm defined by LA(x) = Ax (the matrix product A and x) for each column vector xisinFn

We call LA a left-multiplication transformation

令 A 是 mtimesn 的矩陣所有元素均來自 FLA為 Fn映至 Fm的線性轉換並定義為

LA(x) = Ax其中 x 為行向量（Column vector）且 xisinFnLA被稱為左乘法轉換


8

The next theorem extends this fact to any matrix representation of any linear

transformation defined on finite-dimensional vector spaces

Theorem 33 Let T V W be a linear transformation between finite dimensional vector spaces and

let β and γ be ordered bases for V and W respective Then rank(T) = rank( γβ[T] )

令 T V W且 β 與 γ 分別為有限維度向量空間 V 與 W 的有序基底則

rank(T) = rank( γβ[T] )

改寫 Definition 212

Suppose that Fn and Fm are finite-dimensional vector spaces with ordered bases β =

v1 v2hellip vn and γ = w1 w2hellip wn respectively Let LA Fn Fm be linear and defined

by LA(x) = Ax Then for each j 1≦j≦n there exist unique scalars aijisinF 1≦i≦m such

that sum=

=m

1iiijjA wa)v(L for 1≦j≦n We call the mtimesn matrix A defined by Aij = aij the matrix

representation of LA in the ordered bases β and γ and write γβ= ][LA A

Now that the problem of finding the rank of a linear transformation has been reduced to

the problem of finding the rank of a matrix we need a result that allows us to perform rank-

preserving operations on matrices The next theorem and its corollary tell us how to do this

線性轉換的 Rank 可以被簡化為找矩陣的 Rank如何找矩陣的 Rank

Theorem 34 Let A be an mtimesn matrix If P and Q are invertible mtimesm and ntimesn matrices respectively

then

令 A 為 mtimesn 的矩陣若 P 與 Q 分別為可逆的 mtimesm 與 ntimesn 矩陣則

(a) rank(AQ) = rank(A)

(b) rank(PA) = rank(A)

(c) rank(PAQ) = rank(A)

【Proof】

R(LAQ) = R(LALQ) = LALQ(Fn) = LA(LQ(Fn)) = LA(Fn) = R(LA)

Since LQ is onto rank(AQ) = dim(R(LAQ)) = dim(R(LA)) = rank(A)


9

rank(PAQ) = rank(PA) = rank(A)

註Theorem 34 是為後續 Reduced Row Echelon Form of Matrix 做準備一個矩陣經過

基本行運算或基本列運算或基本行與列運算後並不會改變 RANK

Corollary

Elementary row and column operations on a matrix are rank-preserving

矩陣的基本列與行運算不會改變矩陣的 Rank

此推論結果是計算反矩陣解線性方程組的重要依據

Theorem 35 The rank of any matrix equals the maximum number of its linearly independent

columns that is the rank of a matrix is the dimension of the subspace generated by its

column

任一矩陣的 Rank 等於其最大線性獨立行向量數目即矩陣的 Rank 是由矩陣的

行向量生成的子空間的維度【LA Fn Fm】

【Proof】

For any AisinMmtimesn(F)

rank(A) = rank(LA) = dim(R(LA))

Let β be the standard basis of Fn Then β spans Fn and hence by Theorem 22

R(LA ) = span(LA (β)) = span(LA(e1) LA(e2)hellip LA(en))

But for any j we have seen in Theorem 213(b) that LA(ej) = Aej = aj where aj is the

jth column of A Hence R(LA ) = span(a1 a2 an)

Thus rank(A) = rank(LA) = dim(span(a1 a2 an))

對任意矩陣 Arank(A) = rank(LA) = dim(R(LA))

令 β為 Fn 的標準基底且生成 Fn

依據 Theorem 22

LA的值域 Range 可由 LA(β)來生成

R(LA) = span(LA (β)) = span(LA(e1) LA(e2)hellip LA(en))

對任意 j 而言由 Theorem 213(b) 可知 LA(ej) = Aej = aj其中aj 為 A 的第 j

個行（aj 相當於 Theorem 213(b)的 vj 或 Theorem 22 的 T(vj)）因此 R(LA) =


10

span(a1 a2 an)

所以 rank(A) = rank(LA) = dimension of R(LA) = dim(span(a1a2an))

Theorem 22 令 V 與 W 為向量空間且 T V W 為線性轉換若 β = v1 v2hellip vn

是定義域（Domain）V 的基底則 T 的值域 Range R(T)可由 T(β)來生成意即 R(T)

= span(T(β)) = span(T(v1) T(v2)hellip T(vn))

Theorem 213 A 為 mtimesn 的矩陣與 B 為 ntimesp 的矩陣uj 與 vj分別為 AB 與 B 的第 j 行

（1 j p≦ ≦ ）則

(a) uj = A vj

(b) vj = B ej where ej is the jth standard vector of Fp

EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A

Observe that the first and second columns of A are linearly independent and that the

third column is a linear combination of the first two Thus

A 的第一行與第二行是線性獨立第三行則是前兩行的線性組合於是

rank(A) = dim⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

In order to compute the rank of a matrix A it is frequently useful to postpone the use of

Theorem 35 until A has been suitable modified by means of appropriate elementary row and

column operations so that the number of linearly independent columns is obvious

為了利用 Theorem 35 求矩陣 A 的 Rank（等於最大線性獨立行向量數目）A 可

先經過基本列或行運算使其線性獨立的行數變得比較明顯

EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A

If we subtract the first row of A from rows 2 and 3 (type 3 elementary row operations)

the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121

If we now substract twice the first column from the second and substract the first

column from the third (type 3 elementary column operations) we obtain

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001

（後二行線性相依最大線性獨立行數為 2）

It is now obvious that the maximum number of linearly independent column of this matrix is 2 Hence the rank of A is 2

Theorem 36 Let A be an mtimesn matrix of rank r Then r m r n and by means of a finite number ≦ ≦

of elementary row and column operations A can be transformed into the matrix

D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI

where O1 O2 and O3 are zero matrices Thus Dii = 1 for i r and ≦

Dij = 0 otherwise

令 A 為 Rank 等於 r 的 mtimesn 矩陣則 r m≦ 且 r n≦ 經過有限次數的基本行與列

運算讓 A 化成 D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI

其中Ir 為對角元素為 1 的對角矩陣即 A 可經

過有限次數的基本行及列運算取得沿對角線前 r 個位置皆為 1其餘各處皆為 0 的

矩陣 D

【Proof】

If A is the zero matrix r = 0 In this case the conclusion follows with D = A

若 A 為零矩陣則 r = 0 且 D = A

Now suppose that AneO and r = rank(A) then r gt 0

The proof is by mathematical induction on m the number of rows of A

若 A 非零矩陣且 r = rank(A)則 r gt 0


12

採用數學歸納法

Suppose that m = 1

A can be transformed into the matrix (1 0 hellip0)

So rank(D) = rank(A) = 1 by Corollary to Theorem 34 and Theorem 35 Thus the

theorem is established for m = 1

設 m = 1則 A 可以化成(1 0 hellip0)

依據 Corollary to Theorem 34 及 Theorem 35

rank(D) = rank(A) = 1

故本定理對 m = 1 的矩陣是成立的

Next assume that the theorem holds for any matrix with at most m - 1 row (for some m

gt1)

We must prove that the theorem holds for any matrix with m rows

其次假設本定理對至多含 m - 1 列的矩陣皆成立（m gt 1）

至於含 m 列的矩陣是否成立

If n = 1 Theorem 36 can be established in a manner analogous to that for m = 1

若 n = 1則可按 m = 1 的方式證明本定理對於 n = 1 的矩陣成立

We now suppose that n gt 1 Since A ne O Aijne0 for some i j By means of at most one

elementary row and at most one elementary column operation (each of type 1) We can

move the nonzero entry to the 11 position By means of of at most one additional type 2

operation we can assure a 1 in the 11 position By means of at most m ndash 1 type 3 row

operations and at most n ndash 1 type 3 column operations we can eliminate all nonzero entries

in the first row and the first column with the exception of the 1 in the 11 postion

Thus with a finite number of elementary operation A can be transformed into a

matrix

經過有限次數的基本行及列運算可將 A 轉換成

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001

B where Brsquo is an (m-1)times(n-1) matrix

Brsquo has rank one less than B

Brsquo的 Rank 比 B 少 1

Since rank(A) = rank(B) = r rank(Brsquo) = r ndash 1 Therefore r ndash1 ≦ m ndash 1 and r ndash1 ≦


13

n ndash 1 by the induction hypothesis Hence r ≦ m and r ≦ n

由於 rank(A) = rank(B) = r故 rank(Brsquo) = r ndash 1

依數學歸納法 r ndash1 ≦ m ndash 1 且 r ndash1 ≦ n ndash 1

因此 r ≦ m 且 r ≦ n

Also by the induction hypothesis Brsquo can be transformed by a finite number of

elementary row and column operations into the (m-1)times(n-1) matrix Drsquo such that

同樣地依據數學歸納法Brsquo 可經過有限次數的基本行與列運算轉換成 (m-

1)times(n-1) 的矩陣 Drsquo使得

Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI

where O4 O5 and O6 are zero matrices

That is Drsquo consists of all zeros except for its first r-1 diagonal entries which are ones

Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D

We see that the theorem now follows once we show that D can be obtained from B by

means of a finite number of elementary row and column operations

D 可由 B 經過有限次基本行及列運算而取得

Thus since A can be transformed into B and B can be transformed into D each by a

finite number of elementary operations A can be transformed into D by a finite number of

elementary operations

經過有限次基本行及列運算A B D

Finally since Drsquo contains ones as its first r ndash 1 diagonal entries D contains ones as its

first r diagonal entries and zeros elsewhere

最後Drsquo沿對角線前 r-1 個位置皆為 1故 D 沿對角線前 r 個位置皆為 1其餘

各處皆為 0

EXAMPLE 3 Consider the maxtrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14

By means of a succession of elementary row and column operations we can transform

A into a matrix D as in Theorem 36

透過一系列的基本行與列運算將 A 轉換成 D

D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

By the corollary to Theorem 34 rank(A) = rank(D) Clearly however rank(D) = 3 so rank(A) = 3

Corollary 1 Let A be an mtimesn matrix of rank r Then there exist invertible matrices B and C of sizes

mtimesm and ntimesn respectively such that D = BAC where D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI

is the mtimesn matrix

in which O1 O2 and O3 are zero matrices

令 A 為 mtimesn 的矩陣Rank 為 r存在可逆的 mtimesm 矩陣 B 與 ntimesn 矩陣 C使得

D = BAC其中D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI

為 mtimesn 的矩陣O1O2 及 O3 為零矩陣

【Proof】

By Theorem 36 A can be transformed by means of a finite number of elementary

row and column operations into the matrix D

We can appeal to Theorem 31 each time we perform an elementary operation Thus

there exist elementary mtimesm matrices E1 E2hellipEp and elementary ntimesn matrices G1

G2hellipGq such that D = EpEp-1hellipE2E1AG1G2hellipGq

By Theorem 32 each Ej and Gj is invertible Let B = EpEp-1hellipE2E1 and C =

G1G2hellipGq Then B and C are invertible and D = BAC

依據 Theorem 36A 可經過有限次的基本列與行運算化成 D

引用 Theorem 31每一次皆實施某一個基本列運算於是存在基本的 mtimesm 矩

陣 E1 E2hellip Ep 與基本的 ntimesn 矩陣 G1 G2hellip Gq 使得 D = EpEp-

1hellipE2E1AG1G2hellipGq

依據 Theorem 32 每一個基本矩陣 Ej與 Gj 皆為可逆令 B = EpEp-1hellipE2E1 與 C =

G1G2hellipGq於是 B 與 C 皆為可逆且 D = EpEp-1hellipE2E1AG1G2hellipGq = BAC


15

Theorem 31 令 AisinMmtimesn(F)並假設 B 是由 A 經實施某一基本列【或行】運算後所得

到的矩陣則存在一個 mtimesm 【或 ntimesn】基本矩陣 E使得 B = EA【或 B = AE】E

矩陣是由 Im 【或 In】經實施與「由 A 得到 B」相同的基本列【或行】運算後所得

到的矩陣反之若 E 是一個 mtimesm【或 ntimesn】的基本矩陣則 EA【或 AE】是由 A

經過與「由 Im 【或 In】得到 E」相同的基本列【或行】運算後所得到的矩陣

Theorem 32 基本矩陣為可逆矩陣且基本矩陣的反矩陣亦是同一型的基本矩陣

Theorem 36 令 A 為 Rank 等於 r 的 mtimesn 矩陣則 r m≦ 且 r n≦ 經過有限次數的基

本行與列運算讓 A 化成 D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI

其中Ir為對角元素為 1 的對角矩陣

Corollary 2 Let A be an mtimesn matrix Then

(a) rank(At) = rank(A)

矩陣轉置後的 RANK 不變

(b) The rank of any matrix equals the maximum number of its linearly independent

rows that is the rank of a matrix is the dimension of the subspace generated by

rows

任一矩陣的 Rank 等於最大線性獨立列向量的數目即矩陣的 Rank 等於由該

矩陣列向量所生成的子空間的維度

(c) The rows and columns of any matrix generate subspaces of the same dimension

numerically equal to the rank of the matrix

任一矩陣的列向量與行向量生成的子空間的維度相同維度等於該矩陣的

Rank

【Proof】

By Corollary 1 there exist invertible matrices B and C such that D = BAC where D

satisfies the stated conditions of the corollary Taking transposes we have

Dt = (BAC) t = CtAtBt

Since B and C are invertible so are Bt and Ct

Hence by Theorem 34 rank(At) = rank(CtAtBt) = rank(Dt)

Suppose that r = rank(A)

Then Dt is an ntimesm matrix with the form of the matrix D in Corollary 1 and hence


16

rank(Dt) = r by Theorem 35

Thus

rank(At) = rank(Dt) = r = rank(A)

依據 Corollary 1存在可逆矩陣 B 與 C使得 D = BAC

取其轉置可得Dt = (BAC) t = CtAtBt

由於 B 與 C 為可逆故 Bt 與 Ct 亦為可逆

依據 Theorem 34得知 rank(At) = rank(CtAtBt) = rank(Dt)

設 r = rank(A)

則 Dt為一 ntimesm 的矩陣型式與 D 相同（參考 Corollary 1）

且依據 Theorem 35 得知 rank(Dt) = r

於是 rank(At) = rank(Dt) = r = rank(A)

Theorem 34 Let A be an mtimesn matrix If P and Q are invertible mtimesm and ntimesn matrices

respectively then 令 A 為 mtimesn 的矩陣若 P 與 Q 分別為可逆的 mtimesm 與 ntimesn 矩陣

則




Theorem 35 The rank of any matrix equals the maximum number of its linearly

independent columns that is the rank of a matrix is the dimension of the subspace

generated by its column 任一矩陣的 Rank 等於其最大線性獨立行向量數目即矩陣

的 Rank 是由矩陣的行向量生成的子空間的維度【LA Fn Fm】

Corollary 3 Every invertible matrix is a product of elementary matrices

每一可逆矩陣均可為基本矩陣的乘積

【Proof】

If A is an invertible ntimesn matrix then rank(A) = n Hence the matrix D in Corollary 1

equals In and there exist invertible matrices B and C such that In = BAC

As in the proof of Corollary 1 note that B = EpEp-1hellipE2E1 and C = G1G2hellipGq where

the Eirsquos and Girsquos are elementary matrices Thus A = B-1InC-1 = B-1C-1 so that

A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17

The inverses of elementary matrices are elementary matrices however and hence A is

the product of elementary matrices

若 A 為可逆的 ntimesn 矩陣則 rank(A) = n

因此 Corollary 1 中的矩陣 D 為 In且存在可逆的 B 與 C 矩陣使得 In = BAC

如同 Corollary 1 的證明一樣B = EpEp-1hellipE2E1 且 C = G1G2hellipGq其中Eirsquos 與

Girsquos 皆為基本矩陣

於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1

基本矩陣的反矩陣亦為基本矩陣A 是基本矩陣的乘積

Theorem 37 Let T V W and U W Z be linear transformations on finite-dimensional vector

spaces V W and Z and let A and B be matrices such that the product AB is defined Then

令 VW 與 Z 為有限維度的向量空間T V W 與 U W Z 為線性轉換且

A 與 B 為滿足乘積 AB 定義的矩陣則

(a) rank(UT) ≦ rank(U)

(b) rank(UT) ≦ rank(T)

(c) rank(AB) ≦ rank(A)

(d) rank(AB) ≦ rank(B)

【Proof】

(a) Clearly R(T)subeW

Hence R(UT) = UT(V) = U(R(T)) sube U(W) = R(U)

Thus rank(UT) = dim(R(UT)) ≦ dim (R(U)) = rank(U)

(c) By (a) rank(AB) = rank(LAB) = rank (LALB) ≦ rank(LA) = rank(A)

(d) By (c) and Corollary 2 to Theorem 36

rank(AB) = rank((AB)t) = rank(B t A t) ≦ rank(B t) = rank (B) (b) Let α β and γ be ordered bases for V W and Z respectively and let Arsquo = γ

β]U[

and Brsquo = βα]T[ Then ArsquoBrsquo = γ

α]UT[ by Theorem 211

Hence by Theorem 33 and (d)

rank(UT) = rank(ArsquoBrsquo) ≦ rank(Brsquo) = rank (T)

Theorem 211 Let V W and Z be finite-dimensional vector space with ordered bases α β

and γ respectively Let T V W and U W Z be linear transformation Then


18

βα

γβ

γα = [T][U][UT] 令 VW 與 Z 是有限維度的向量空間αβ 與 γ 分別為 VW 與

Z 的有序基底令 T V W （先 α β）且 U W Z （後 β γ）則 βα

γβ

γα = [T][U][UT]

Theorem 33 Let T V W be a linear transformation between finite dimensional vector

spaces and let β and γ be ordered bases for V and W respective Then rank(T) = rank( γ

β[T] ) 令 T V W且 β 與 γ 分別為有限維度向量空間 V 與 W 的有序基底

則 rank(T) = rank( γβ[T] )

EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121

A （第一列與第二列是 LI） rank(A) = 2

(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131

A （第三列與第一二列是 LD）rank(A) = 2

(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321

A （三列為 LI） rank(A) = 3

The inverse of a Matrix

We have remarked that an ntimesn matrix is invertible if and only if its rank is n Since we

know how to compute the rank of any matrix we can always test a matrix to determine

whether it is invertible We now provide a simple technique for computing the inverse of a

matrix that utilizes elementary row operations

已知 ntimesn 矩陣為可逆的『若且唯若』條件為該矩陣的 Rank 等於 n

DEFINITION 34 Augmented Matrix Let A and B be mtimesn and mtimesp matrices respectively By the augmented matrix )BA(

we mean that the mtimes(n+p) matrix )BA( that is the matrix whose first n columns are the

columns of A and whose last p columns are the columns of B 設 A 與 B 分別為 mtimesn 與 mtimesp 的矩陣則增廣矩陣（Augmented matrix） )BA(


19

為一 mtimes(n+p)的矩陣 )BA( 該矩陣的前 n 行（Column）為 A 的所有的行後 p

行為 B 的所有行

Let A be an invertible ntimesn matrix and consider the ntimes2n augmented matrix C= )IA( n

C= )IA( n )AI()IAAA(CA 1

nn111 minusminusminusminus ==

By Corollary 3 to Theorem 36 A-1 is the product of elementary matrices say

A = EpEp-1hellipE2E1 （A-1 是多個基本矩陣的乘積）

Thus )AI()IAAA(CA 1

nn111 minusminusminusminus == )AI(CA)IA(EEE 1

n1

n11ppminusminus

minus ==

We have the following result If A is an invertible invertible ntimesn matrix then it is

possible to transform the matrix (AIn) into the matrix (InA-1) by means of a finite

number of elementary row operations

若 A 為一可逆的 ntimesn 矩陣則可經過有限次數的基本列運算將矩陣 (AIn) 化成

(InA-1)

Conversely suppose that A is invertible and that for some ntimesn matrix B the matrix

(AIn) can be transformed into the matrix (InB) by a finite number of elementary row

operations Let E1 E2hellip Ep be the elementary matrices associated with these elementary row

operations as in Theorem 31 then

反之假設 A 是可逆且矩陣(AIn)可經有限次數的基本列運算轉換成矩陣(InB)

並令 E1 E2hellip Ep 分別為 Theorem 31 所稱的基本矩陣則

)BI()IA(EEE nn11pp =minus

Let M = EpEp-1hellipE2E1 we have 令 M = EpEp-1hellipE2E1

)BI()IA(EEE nn11pp =minus )BI()IA(M)MMA( nn ==


20

Hence MA = In and M = B It follows that M = A-1 So B = A-1

可由 MA = In 及 M = B 得知M 就是 A-1故 B 就是 A-1

Thus we have the follow resultsIf A is an invertible ntimesn matrix and the matrix

(AIn) is transformed in a matrix of the form (InB) by means of a finite number of

elementary row operations then B=A-1

若 A 為一可逆的 ntimesn 矩陣且矩陣 (AIn) 可經有限次數的基本列運算化成

(InB)則 B=A-1

If on the other hand A is an ntimesn matrix that is not invertible then rank(A) lt n Hence

any attempt to transform (AIn) into a matrix of the form (InB) by means of elementary row

operations must fail

反之若 A 是一不可逆的 ntimesn 矩陣則 rank(A) lt n因此任何想利用基本列運算

要將矩陣 (AIn) 化成 (InB)的努力必定失敗

Thus we have the follow results If A is an notinvertible ntimesn matrix then any attempt

to transform (AIn) into a matrix of the form (InB) produce a row whose first n entries

are zeros

若 A 是一不可逆的 ntimesn 矩陣任何想將矩陣 (AIn) 化成 (InB)的努力最後會

產生一前 n 個元素均為 0 的列

EXAMPLE 5

We determine whether the matrix ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A is invertible and if it is we

compute its inverse

A 是否為可逆矩陣

We attempt to use elementary row operations to transfer


21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420

)IA( into a matrix of the form (InB)

看看能否利用基本行與列運算將矩陣 (AIn) 化成 (InB)

Interchanging rows 1 and 2

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242

Multiplying the first row by 12

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1

Adding -3 times row 1 to row 3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus 1000100

230420121

23

21

Multiplying row 2 by 12 to obtain 1 in the 22 position

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21

Adding -2 times row 2 to row 1 and 3 times row 2 to row 3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21

Multiplying row 3 by 14

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301

Adding appropriate multiply of row 3 to row 1 and 2 completes the process

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成

Thus A is invertible and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+minus+minus+minus+minus+

=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112

121A is invertible and if it is we

compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112

121)IA( into a matrix of the form (InB)


Adding -2 times row 1 to row 2 and -1 times row 1 to row 3 then addig row 2 to row

3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminusminus

101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121

is a matrix with row whose first 3 entries are zeros Therefore A is not invertible

最後一列前 3 個元素均為 0

EXAMPLE 7 Let T P2(R) P2(R) be defined by T(f(x)) = f(x)+frsquo(x)+frdquo(x) where frsquo(x) and frdquo(x)

denote the first and second derivatives of f(x) We use Corollary 1 to Theorem 218 to test

T for invertibility and compute the inverse if T is invertible Taking β=1 x x2 to be the

standard ordered basis of P2(R) we have

令 T P2(R) P2(R) 為線性轉換且定義為 T(f(x)) = f(x)+frsquo(x)+frdquo(x)利用

Corollary 1 to Theorem 218 來測試 T 是否可逆並計算其逆轉換取 β=1 x x2為

P2(R)的標準有序基底

T(1) = 1 = 1bull1+0bullx+0bullx2

T(x) = 1+x = 1bull1+1bullx+0bullx2


23

T(x2) = 2+2x+x2 = 2bull1+2bullx+1bullx2

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211

[T] 稱為 T 相對有序基底 β的座標向量

依據 Corollary 1 to Theorem 218 T 可逆『若且唯若』 β[T] 可逆且

1-1 )[T](][T minusββ =

Using the method of Examples 5 and 6 we can show that β[T] is invertible with

inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1

Thus T is is invertible and βminusminus

β = ][T)([T] 11 Hence by Theorem 214 we have

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T

Therefore T-1(a0+a1x+a2x2) = (a0-a1)+(a1-2a2)x+a2x2 Corollary 1 to Theorem 218 Let V be finite-dimensional vector space with an ordered basis β and let T V V be linear Then T is invertible if and only if β[T] is invertible

Furthermore 1-1 )[T](][T minusββ = V 為有限維度的向量空間β 是 V 的有序基底令 T

V V 為線性轉換 T 為可逆的『若且唯若』條件為 β[T] 可逆再者


Theorem 214 Let V and W be finite-dimensional vector spaces having ordered bases β and γ respectively and let T V W be linear Then for each uisinV we have β

γβγ = [u][T][T(u)]

V 與 W 為有限維度的向量空間β 與 γ分別 V 與 W 的有序基底令 T V W 為線

性轉換則 V 中的 u 映至 W 的結果 T(u) 相對於有序基底 γ 的座標向量為

βγβγ = [u][T][T(u)]

Invertible and Inverse of MATRIX Let A be an n times n matrix If a matrix B can be found such that AB = BA = In then A is


24

said to be invertible and B is called the inverse of A (denote B = A-1 and A-k=(A-1)k )

If such a matrix B does not exist then A has no inverse

EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=

Thus AB = BA = I2 proving that the matrix A has inverse B

Theorem The inverse of an invertible matrix is unique

Let B and C be inverse of A

Thus AB = BA = In and AC = CA = In

Multiply both sides of the equation AB = In by C

C(AB) = CIn (CA)B = C

InB = C B = C

Thus an invertible matrix has only one inverse

How to find AHow to find A--11

Based on the GaussBased on the Gauss--Jordan algorithmJordan algorithmLet A be an invertible matrix Then AALet A be an invertible matrix Then AA--11=I=Inn

Let the columns of Let the columns of AA--11 be be XX11 XX22 helliphellip XXnn and the columns of I and the columns of Inn be be CC11CC22helliphellipCCnn

Find AFind A--11 by finding by finding XX11 XX22 helliphellip XXnn Write the equation AAWrite the equation AA--11=I=Inn in the formin the form

[ ] [ ]n21nn211 CCCI XXXALet LL ==minus

[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=

nn2211 CAXCAX CAX === LThus


Solving these systems by using Gauss-Jordan elimination on the large augmented matrix [AC1 C2hellipCn]

Thus when A-1 exits

[ ] [ ] [ ] 1n

1nn ABwhereBIAIIA minusminus ==asympasympL

If the reduced echelon form of [If the reduced echelon form of [AIAInn] is computed and ] is computed and the first part is not of the form Ithe first part is not of the form Inn then A has no inversethen A has no inverse

[ ][ ]n21n

n21

XXXI CCCAmatrix augmented

LL

L

asympasymp


25

Example Example 1212

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminusminusminus

=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusminusminusasymp

101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211

ADetermine the inverse of


⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=minus

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1

ExampleExampleDetermine the inverse of

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2

There is no need to proceed further There is no need to proceed further The reduced echelon form cannot have a one in the (3 3) locatioThe reduced echelon form cannot have a one in the (3 3) location n

The reduced echelon form cannot be of the form [IThe reduced echelon form cannot be of the form [Inn B] B] Thus Thus AAndashndash11 does not existdoes not exist

Properties of Matrix InverseProperties of Matrix Inverse

Let Let AA and and BB be invertible matrices andbe invertible matrices and cc a nonzero scalar a nonzero scalar ThenThen

AA =minusminus 11)( 111 1)( 2 minusminus = A

ccA

111)( 3 minusminusminus = ABAB

nn AA )()( 4 11 minusminus =tt AA )()( 5 11 minusminus =

Color Models Color Models 1212

A color model in the context of graphics is a method of A color model in the context of graphics is a method of implementing colorsimplementing colorsThe common used models include RGB model and YIQ modelThe common used models include RGB model and YIQ modelThe RGB model is used in computer monitors and the YIQ model The RGB model is used in computer monitors and the YIQ model is used in television screensis used in television screensAn RGB computer signal can be converted to a YIQ television An RGB computer signal can be converted to a YIQ television signal using the matrix transformationsignal using the matrix transformation

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB


A signal is converted from a television screen to a computer monA signal is converted from a television screen to a computer monitor itor using inverse of the above matrixusing inverse of the above matrix

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1

CryptographyCryptography 密碼學密碼學 1212

Cryptography is the process of coding and decoding messagesCryptography is the process of coding and decoding messagesThe technique can be traced back to the ancient Greeks Today The technique can be traced back to the ancient Greeks Today governments use sophisticated methods of coding and decoding governments use sophisticated methods of coding and decoding messagesmessagesOne type of code that is extremely difficult to break makes use One type of code that is extremely difficult to break makes use of a of a large matrix to encode a message large matrix to encode a message The receiver of the message decodes it using the inverse of the The receiver of the message decodes it using the inverse of the matrixmatrixThe first matrix is called the encoding matrix and its inverse The first matrix is called the encoding matrix and its inverse is is called the decoding matrixcalled the decoding matrix

Cryptography Cryptography 密碼學密碼學 2222

A A-1Message

Encoding MatrixEncoding Matrix

Decoded MessageTransmission

Decoding MatrixDecoding Matrix


26


Let the message be Let the message be BUY IBM STOCKBUY IBM STOCKAnd the encoding matrix beAnd the encoding matrix be

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433

B U Y B U Y -- I B M I B M -- S T O C KS T O C K2 21 25 27 9 2 13 27 19 20 15 3 112 21 25 27 9 2 13 27 19 20 15 3 11

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding


Let the message be Let the message be BUY IBM STOCKBUY IBM STOCKAnd the decoding matrix beAnd the decoding matrix be

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 334344101


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminusminusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding

3-3 Systems of Linear Equations ndash Theoretical Aspects

The system of equations

mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)

were aij and bi (1≦i≦m and 1≦j≦n) are scalars in a field F and x1 x2hellipxn are n

variables taking values in F is called a system of m linear equations in n unknows over the

field F

上式稱為佈於 Field F 含有 n 個未知數m 個線性方程式的系統或方程組（system

of m linear equations in n unknows）

The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa

A is called the coefficient matrix of the

system (S)（係數矩陣） If we let

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb

b Then the system (S) may

be rewritten as a single matrix equation Ax = b（單一矩陣方程式）

To exploit the results that we have developed we often consider a system of linear

equations as a single matrix equation


27

A solution to the system (S) is an n-tuple n

n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

= such that As = b The set of

all solutions to the system (S) is called the solution set of the system（解集合） System (S)

is called consistent（相容） if its solution set is nonempty otherwise it is called

inconsistent（不相容）

方程組（S）的解為一 n 序組該序組可以使 As = b方程組的所有解所成的集合

稱為該方程組的解集合（Solution set）方程組的解集合非空集合者稱為「相容」解

集合空集合者稱為「不相容」

EXAMPLE 1

Consider 6x2xx1xx3x2

321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1

（以矩陣表示）

This system has many solutions such as ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1

顯然地方程組無解

We begin our study of system of linear equations by examining the class of

homogeneous system of linear equations 先討論齊次方程組

DEFINITION 35 Homogeneous amp Nonhomogeneous

A system Ax=b of m linear equations in n unknowns is said to be homogeneous if b=0

Otherwise the system is said to be nonhomogeneous

方程組 Ax = bb = 0 稱為齊次方程組否則稱為非齊次方程組

Any homogeneous system has at least one solution namely the zero vector The next

result gives further information about the set of solutions to a homogeneous system


28

齊次方程組至少有一個解那就是『零向量』

Theorem 38 Let Ax = 0 be a homogeneous system of m linear equations in n unknowns over a field

F Let K denote the set of all solutions to Ax = 0 Then K = N(LA) hence K is a subspace

of Fn of dimension n - rank(LA) = n- rank(A)

令 Ax = 0 代表一佈於 Field 含有 n 個未知數m 個線性方程式的方程組若令

K 代表該方程組的解集合（Solution set）則 K = N(LA)K 為 Fn 內的子空間且其維

度為 n ndash rank (LA) = n - rank(A)

【Proof】

Clearly K = sisinFn As = 0 = N(LA) The second part now follows from the

dimension theorem

Nullity(T)+rank(T) = dim(V)

Nullity (LA) = dim (Fn) ndashrank (LA ) = n ndash rank (A)

DEFINTION 27 Let V and W be vector space and let T V W be linear We define

the null space (or kernel) N(T) of T to be the set of all vectors x in V such that T(x) = 0

that is N(T) = xisinV T(x) = 0 V 與 W 為向量空間且 T V W 為線性轉換定義

線性轉換 T 的 Null space 為 V 內所有滿足 T(x) = 0 的向量 x 所形成的集合註記為

N(T)Null space 的元素 x 經線性轉換 T 轉換後所對應的「像」為 0意即 T(x) = 0

Null space 內的元素的「像」皆為 0

DEFINITION 217 Let A be an mtimesn matrix with entries from a field F We denote by LA

the mapping LA Fn Fm defined by LA(x) = Ax (the matrix product A and x) for each

column vector xisinFn We call LA a left-multiplication transformation 令 A 是 mtimesn

的矩陣所有元素均來自 FLA 為 Fn 映至 Fm的線性轉換並定義為 LA(x) = Ax其中

x 為行向量（Column vector）且 xisinFnLA被稱為左乘法轉換

Theorem 23 (Dimension Theorem) Let V and W be vector spaces and let T V W be

linear If V is finite-dimensional then Nullity(T) + rank(T) = dim(V) T 的核次數

（Nullity）與秩（Rank）的和等於定義域的維度 dim(V)

Corollary Let Ax = 0 be a homogeneous system of m linear equations in n unknowns over a field


29

F If m lt n the system Ax = 0 has a nonzero solution

令 Ax = 0 代表一佈於 Field 含有 n 個未知數m 個線性方程式的方程組若 m

lt n則該方程組具有一非零解

【Proof】

Suppose that m lt n Then rank(A) = rank (LA)≦m Hence dim(K) = n - rank(LA) ≧

n ndash m gt 0 where K = N(LA) Since dim(K) gt 0 Kne 0 Thus there exists a nonzero

vector sisinK so s is a nonzero solution to Ax = 0

設 m lt n則 rank(A) = rank (LA) ≦ m依據 Theorem 38K = sisinFn As = 0

= N(LA) = dim(K) = n - rank(LA) ≧ n ndash m gt 0表示 dim(K) gt 0Kne 0故存在非

零向量 s使得 Ax = 0

EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111

121A be the coefficient matrix of this system It is clear that rank(A)

= 2 If K is the solution set of this system then dim(K) = 3-2 = 1 Thus any nonzero

solution constitutes a basis for K

若 K 是方程組的解所組成的集合（簡稱為解集合）依 Theorem 38 得知 dim(K)

= 3 - 2 = 1因此任一非零解將形成 K 的基底

For example since ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a solution to the given system

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for

K Thus any vector in K is of the form ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1是方程組的一個解故

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 為 K 的基底K 內的任一向量均

具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30

We now turn to the study of nonhomogeneous systems Our next result shows that the

solution set of a nonhomogeneous system Ax = b can be described in terms of the solution of

the homogeneous system Ax = 0 We refer to the equation Ax = 0 as the homogeneous system

corresponding to Ax = b

以下將進一步探討「非齊次方程組」非齊次方程組 Ax = b 的解集合可利用齊次方

程組 Ax = 0 的解來加以說明因此我們可將 Ax = 0 視為 Ax = b 的齊次方程組

Theorem 39 Let K be the solution set of a system of linear equations Ax = b and let KH be the

solution set of the corresponding homogeneous system Ax = 0 Then for any solution s to

Ax = b K =s + KH = s+k kisinKH

設 K 為線性方程組 Ax = b 的解集合且 KH為對應的齊次方程組的解集合則

對 Ax = b 的任一個解 s 而言K =s + KH = s+k kisinKH

【Proof】

Let s be any solution to Ax = b

We must show that K =s + KH

If wisinK then Aw = b Hence

A(w - s) = Aw ndash As = b ndash b = 0

So w - sisinKH

Thus there exists kisinKH such that w - s = k

It follows that w = s + k isin s +KH and therefore Ksube s +KH

Conversely suppose that wisins +KH then w = s + k for some kisinKH

But then Aw = A(s + k) = As + Ak = b + 0 = b so wisinK

Therefore s +KHsubeK and thus K=s +KH

令 s 為 Ax = b 的任一個解我們必須證明「K =s + KH」

若 wisinK則 Aw = b（即 w 為 Ax = b 的解）

於是 A(w - s) = Aw ndash As = b ndash b = 0

所以 w - sisinKH（即 w-s 為 Ax = 0 的解）

w - s 既然是 KH的元素表示有一元素 kisinKH使得 w - s = kw = s + k

w = s + kisins +KH


31

因此 Ksube s +KH

反之假設 wisins +KH則 KH中存在一元素 k（kisinKH）使得 w = s + k

因 Aw = A(s + k) = As + Ak = b + 0 = b

故 wisinK 是 Ax = b 的解

因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++

The corresponding homogeneous system is the system in Example 2 It is easily

verified that ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411

s is a solution to the preceeding nonhomogeneous system So the

solution set of the system is ⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411

s 是非齊次方程組 Ax = b 的解

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1是齊次方程組的解

依據 Theorem 39 得知線性方程組 Ax = b 的解集合⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K

The following theorem will provide us with a means of computing solutions to certain

system of linear equations

Theorem 310~311 提供計算線性方程組的解的方法

Theorem 310 Let Ax = b be a system of n linear equations in n unknowns If A is invertible then the


32

system has exactly one solution namely A-1b Conversely if the system has exactly one

solution then A is invertible

令 Ax = b 為含 n 個未知數n 個線性方程式的方程組若 A 為可逆則方程組

恰有一個解解為 A-1b反之若該方程組恰有一個解則 A 為可逆

【Proof】

Suppose that A is invertible

Substituting A-1b into system we have A(A-1b)=( AA-1)b = b

Thus A-1b is a solution

If s is an arbitrary solution then As = b

Multiplying both sides by A-1 gives s= A-1b

Thus the system has one and only one solution namely A-1b

假設 A 為可逆將 A-1b 代入方程組可得 A(A-1b)=( AA-1)b = b

故 A-1b 確為方程組的解

若 s 為方程組的任意解則 As = b

兩邊乘上 A-1可得 s= A-1b

因此方程組僅有唯一的解 A-1b

Conversely suppose that the system has exactly one solution s

Let KH denot the solution set for the corresponding homogeneous system Ax = 0

By Theorem 39 s = s + KH But it is so only if KH = 0 Thus N(LA) = 0 and

hence A is invertible

反之假設方程組恰有一個解 s

令 KH為相對應齊次方程組 Ax = 0 的解集合

依據 Theorem 39要能s = s + KH唯有 KH = 0

KH = 0即表示 N(LA) = 0故 A 為可逆

EXAMPLE 4 Consider the following system of three linear equations in three unknowns

1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+

In Example 5 of Section 32 we compute the inverse of the coefficient matrix A of

this system Thus the system has exactly one solution namely


33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx

Theorem 311 Let Ax = b be a system of linear equations Then the system is consistent （相容）if

and only if rank(A) = rank (A∣b) Matrix (A b) is called the augmented matrix o∣ f the

system Ax = b

令 Ax = b 為線性方程式方程組則此方程組至少有一個解的『若且唯若』條件

為 rank(A) = rank (A∣b) 矩陣(A∣b)稱為方程組 Ax = b 的增廣矩陣

【Proof】

To say that Ax = b has a solution is equivalent to saying that bisinR(LA)

In the proof of Theorem 35 we saw that R(LA ) = span(a1 a2 an) the span of the

column of A

Thus Ax = b has a solution if and only if bisinspan(a1 a2 an)

But bisinspan(a1 a2 an) if and only if span(a1 a2 an) = span(a1 a2 an b)

This last statement is equivalent to

dim (span(a1a2an)) = dim (span(a1a2an b))

So by Theorem 35 the preceeding equation reduces to rank(A) = rank (A∣b)

Ax = b 有一解相當於 bisinR(LA)即 b 屬於線性轉換 LA的 Range

在 Theorem 35 的證明中提到 R(LA ) = span(a1a2an)aj 為 A 的第 j 個行

（Column）即線性轉換 LA 的 Range 係由矩陣 A 的行（Column）所生成

因此 Ax = b 要有一個解的『若且唯若』條件為 bisinspan(a1 a2 an)

至於 bisinspan(a1 a2 an)之條件

bisinspan(a1a2an) 的『若且唯若』條件為 span(a1a2an) = span(a1a2an

b)

其條件相當於


34


依據 Theorem 35該條件可簡化為 rank(A) = rank (A∣b)



generated by its column任一矩陣的 Rank 等於其最大線性獨立行向量數目即矩陣的

Rank 是由矩陣的行向量生成的子空間的維度【LA Fn Fm】

EXAMPLE 5 Recall the system of equations

1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(

rank(A) = 1 and rank (A b) = 2∣

Because the two ranks are unequal the system has no solutions

矩陣 A 與增廣矩陣(A∣b) 的 Rank 不同故方程組沒有解

EXAMPLE 6 We can use Theorem 311 to determine whether (3 3 2) is in the range of the linear

transformation T R3 R3 defined by T(a1 a2 a3) = (a1+a2+a3 a1-a2+a3 a1+a3)

利用 Theorem 311 來判斷 (3 3 2)是否在線性轉換 T 的 Range 中其中T R3

R3 定義為 T(a1 a2 a3) = (a1+a2+a3 a1-a2+a3 a1+a3)

Now (3 3 2)isinR(T) if and only if there exists a vector s = x1 x2 x3 in R3 such that

T(s)=(3 3 2) Such a vector s must be a solution to the system

2xx3xxx3xxx

31

321

321

=+=+minus=++

Since the ranks of the coefficient matrix and the augmented matrix of this system

are 2 and 3 respectively it follows that this system has no solutions Hence (3 3 2) notin

R(T)

(3 3 2)isinR(T) 的『若且唯若』的條件為存在一向量 s = x1 x2 x3isinR3 使得

T(s)=(3 3 2)


35

向量 s 必須是 2xx3xxx3xxx

31

321

321

=+=+minus=++

的解

因係數矩陣⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111

的 Rank 為 2增廣矩陣⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩

者不相等故方程組無解(3 3 2) notin R(T)

Theorem 311 Let Ax = b be a system of linear equations Then the system is consistent

（相容）if and only if rank(A) = rank (A∣b) Matrix (A b) is called the ∣ augmented

matrix of the system Ax = b 令 Ax = b 為線性方程式方程組則此方程組至少有一個

解的『若且唯若』條件為 rank(A) = rank (A∣b) 矩陣(A∣b)稱為方程組 Ax = b 的

增廣矩陣

Application

In 1973 Wassily Leontief won the Nobel prize in economics for his work in developing

a mathematical model that can be used to describe various economic phenomena

1973 年Wassily Leontief 因研究一數學模式來描述各種經濟現象而獲頒諾貝爾經

濟獎

CLOSED MODEL

We begin by considering a simple society composed of three peoples (industries) ndash a

farmer who gows all the food a tailor who makes all the clothing and a carpenter who builds

all the housing We assume that each person sells to and buy from a central pool and that

everything produced is consumed Since no commodities either enter or leave the system this

case is referred to as the CLOSE MODEL

考慮一個單純社會的三種人生產食物的農夫製衣的裁縫師與蓋住屋的木匠

並假設採取集中聯營買賣且每一件產品皆付諸消費由於過程中沒有任何成品進入

或脫離這個系統所以稱為「封閉模式」

Each of these three individual consumes all three of the commodities produced in the


36

society Suppose that the proportion of each of the commodities consumed by each person is

given in the following table

社會中每一位皆消費所生產的三種成品且每一種產品也都被三位所消費其消

費比例如下

食物 Food 衣物 Clothing 住屋 Housing

農夫 Farmer 040 020 020

裁縫師 Tailor 010 070 020

木匠 Carpenter 050 010 060

Let p1 p2 and p3 denote the incomes of the farmer tailor and carpenter respectively To

ensure that this society survives we require that the consumption of each individual equals

his or her income

令 p1p2 與 p3 分別表示農夫裁縫師與木匠的收入並為確保這個封閉社會可以

存活假設每一位的收入等於消費

Note that the farmer consumes 20 of the clothing Because the total cost of all clothing

is p2 the tailorrsquos income the amount spent by the farmer on clothing is 020 p2 Moreover the

amount spent by the farmer on food clothing and housing must equal the farmerrsquos income

and so we obtain the equation 040 p1 + 020 p2 + 030 p3 = p1

對農夫而言農夫的衣物消費佔 20然因所有衣物的消費總額 p2 等於裁縫師的

收入所以農夫在衣物上的消費為 020 p2且農夫在食物衣物與住屋的總消費要等

於農夫的收入 p1所以農夫的收入與消費關係式

040 p1 + 020 p2 + 020 p3 = p1

Similary equations describing the expenditures of the tailor and carpenter produce the

following system of linear equations

同理分別對裁縫師與木匠進行討論可獲得下列方程組


37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3

This system can be written as Ap = p

其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400

A 稱為輸入－輸出（或消費）矩陣

（Input-output (or consumption) matrix）

Ap = p 稱為均衡條件（Equilibrium condition）

能否利用不等式 p ≦ p 來替代等式

At first it may seem reasonable to replace the equilibrium condition by the inequality

Ap ≦ p that is the requirement that consumption not exceed production But in fact Ap ≦ p implies that Ap = p in the closed model

For otherwise there exists a k for which

sumgt

jjkjk pAp

Hence since the columns of A sum to 1

sumsumsumsum ==gt

jj

i jjij

ii ppAp which is a contradiction （出現矛盾）

答案是在封閉系統中Ap ≦ p Ap = p

One solution to the homogeneous system (I ndash A) x = 0 which is equivalent to the

equilibrium condition is ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38

齊次方程組(I ndash A) x = 0 或均衡條件的一個解為 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p

We may interpret this to mean that the society survives if the farmer taior and carpenter

have incomes in the proportions 253540

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p 可以解釋為若農夫裁縫師與木匠收入比例為 253540則社會將

能生存下去

Theorem 312

Let A be an ntimesn input-output matrix having the form ⎟⎟⎠

⎞⎜⎜⎝

⎛=

EDCB

A where D is a 1times(n-

1) positive vector and C is an (n-1)times1 positive vector Then (I-A)x = 0 has a one-

dimensional solution set that is generated by a nonnegative vector

設 A 為 ntimesn 輸入－輸出矩陣 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

EDCB

A 其中D 為 1times(n-1)正向量C 為

(n-1)times1 正向量則(I-A)x = 0 具有一維解集合且該解集合係由一非負向量所生成

Any input-output matrix with all positive entries satisfies the hypothesis of this theorem

任何輸入－輸出矩陣滿足 Theorem 312 的基本假設條件例如

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL

In the OPEN MODEL we assume that there is an outside demand for each of the

commodities produced Returning to our simple society let x1 x2 and x3 be the monetary

values of food clothing and housing produced with respective outside demands d1 d2 and d3

Let A be the 3times3 matrix such that Aij represents the amount of commodity i required to


39

produce one monetary unit of commodity j Then the value of the surplus of food in the

society is x1 ndash (A11x1+A12x2+A13x3) that is the value of food produced minus the value of

food consumed while producing the three commodities The assumption that everything

produced is consumed gives us a similar equilibrium condition for the open model namely

that the surplus of each of the three commodities must equal the corresponding outside

demands Hence i

3

1jjiji dxAx =minussum

=

for j=1 2 3

在「開放模式」中我們假設每一產品皆有外來的需求在前述的單純社會裡

令 x1x2 與 x3 分別代表食物衣物與住屋的貨幣價值d1d2 與 d3 為食物衣物與住

屋的外來需求價值令 A 為 3times3 矩陣Aij 表示生產『一個貨幣單位』的產品 j 所需要

的產品 i 的價值於是此社會中食物的盈餘價值為 x1 ndash (A11x1+A12x2+A13x3)即食物的

價值減去生產食物衣物與住屋所需消耗的食物的價值意即生產出來的食物價值扣

掉用去生產食物衣物與住屋所消耗掉的價值）（A11A12A13 為生產一個貨幣單位

的食物 1衣物 2 與住屋 3 所需消耗的食物 1 數量A12x2 表示生產貨幣價值為 x2的衣物

所需要的食物價值）對每一種產品而言其盈餘價值等於外來的需求價值於是

i

3


=

for j=123

Suppose that 30 cents worth of food 10 cents worth of clothing and 30 cents worth of

housing are required for the production of $1 worth of food Similar suppose that 20 cents

worth of food 40 cents worth of clothing and 20 cents worth of housing are required for the

production of $1 worth of clothing Finally suppose that 30 cents worth of food 10 cents

worth of clothing and 30 cents worth of housing are required for the production of $1 worth

of housing Then the input-output matrix is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A

假設生產價值 1 元的食物需要價值 30 分的食物（A11 = 03）價值 10 分的衣物

（A21 = 01）及價值 30 分的住屋（A31 = 03）生產價值 1 元的衣物需要價值 20 分的

食物（A12 = 02）價值 40 分的衣物（A22 = 04）及價值 20 分的住屋（A32 = 02）生


40

產價值 1 元的住屋需要價值 30 分的食物（A13 = 03）價值 10 分的衣物（A23 = 01）

及價值 30 分的住屋（A33 = 03）於是輸入輸出矩陣 A 為

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A

若食物的外來需求值為 300 億元衣物為 200 億元住屋值為 1 百億元則

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900

x 表示要滿足外來需求必須生產的食物價值為 900 億元衣物價值為

600 億元住屋價值為 700 億元

Theorem

Let AX = Y be a system of n linear equations in n variables If Andash1 exists the

solution is unique and is given by X = Andash1Y

【Proof】

Substitute X = Andash1Y into the matrix equation

AX = A(Andash1Y) = (AAndash1)Y = InY = Y

The solution is unique

Let X1 be any solution thus AX1 = Y Multiplying both sides of this equation by Andash1

gives

Andash1AX1= Andash1Y InX1 = Andash1Y X1 = Andash1Y


41


2x5x3x3x5x3x21x2x x

321

321

321

minus=++minus=minusminus=minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1

Solve the system of equations

This system can be written in the following matrix form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1

If the matrix of coefficients is invertible the unique solutionIf the matrix of coefficients is invertible the unique solution isis


1 2 1 issolution unique The 321 =minus== xxx

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1

The The LeontiefLeontief IO Model in IO Model in Economics Economics 1515

The Leontief input-output model is used to analyze the interdependence of economics Wassily Leontief received a Nobel Prize in 1973 for his work in thie fieldThe practical applications of this model have proliferated until it has now become a standard tool for investigating economic structures ranging from cities and corporation to states and countriesConsider an economics situation that involves ninterdependent industries The output of any one industry is needed as input by other industries and even possibly by the industry itself


How a mathematical model involving a system of linear equations can be constructed to analyze such a situationAssume that each industry produces one commodity Let aij denote the amount of input of a certain commodity i to produce unit output of commodity j In our model let the amounts of input and output be measured in dollars Thus for example a34 = 045 means that 45 centsrsquo worth of commodity 3 is required to produce one dollarrsquos worth of commodity 4


aij = amount of commodity i in one dollar of commodity jaij =生產一元價值的j產品需要耗用的i產品價值

The elements aij called input coefficients define a matrix A called the input-output matrix which describes the interdependence of the industriesExtending the model to include an open sector (all non-producing sectors) The product of industries may go not only into other producing industries but also into other non-producing sectors of the economy such as consumers and governments


di = demand of the open sectors from industry ixi=total output of industry I necessary to meet demands of all n industries and the open sector

sectoropen of demand

nindustry of demand

2industry of demand

22

1industry of demand

11

industry ofoutput total

ininii

i

i dxaxaxax ++sdotsdotsdot++=

output ofportion sectoropen

output ofportion tryinterindusoutput total

DAXX +=Demand matrixDemand matrix

Output matrixOutput matrix


Applying the model to the economy of a country X Applying the model to the economy of a country X represents the total output of each of the producing sectors represents the total output of each of the producing sectors of the economy and AX describes the contributions made of the economy and AX describes the contributions made by the various sectors to fulfilling the intersectional input by the various sectors to fulfilling the intersectional input requirements of the economyrequirements of the economyD is equal to (XD is equal to (X--AX) the difference between total output AX) the difference between total output X and industry transaction AXX and industry transaction AXD is thus the gross national product of the economyD is thus the gross national product of the economy


Consider an economy consisting of three industries having Consider an economy consisting of three industries having the following inputthe following input--output matrix output matrix AA Determine the output Determine the output levels required of the industries to meet the demands of levels required of the industries to meet the demands of the other industries and of the open sector in each casethe other industries and of the open sector in each case

in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA

The unit of The unit of DD are millions of dollarsare millions of dollars


42


We wish to compute the output levels X that correspond to We wish to compute the output levels X that correspond to the various open sector demands D X is given by the the various open sector demands D X is given by the equation X = AX + D Rewritten as followsequation X = AX + D Rewritten as follows

X ndash AX = D rArr (I ndash A)X = D rArr X = (I ndash A) ndash1 D

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI

output ofingcorrespond valuesvarious

4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus

minusuarruarruarruarruarruarruarr

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=


By using Gauss-Jordan elimination

dollars of milliond are units The

40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


The output levels necessary to meet the demandsThe output levels necessary to meet the demands

IAAAIAI m =++++minus ))(( 2 L

mAAAIAI ++++=minus minus L21)(

Computing (IComputing (I--A) A) --11

1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(

+minus=++++minus++++=

++++minusLL

L

NOTE 0A 1m rarr+

Population Movement Population Movement 1313

In 2000 58 million of people live in cities and 142 million of In 2000 58 million of people live in cities and 142 million of people people live in the surrounding suburbs Represent this information by tlive in the surrounding suburbs Represent this information by the he matrix matrix

The probability of a person staying in the city was 096 Thus tThe probability of a person staying in the city was 096 Thus the he probability of moving to the suburbs was 004 probability of moving to the suburbs was 004 The probability of a person moving to the city was 001 the The probability of a person moving to the city was 001 the probability of remaining in suburbia was 099probability of remaining in suburbia was 099

suburbcity

990040010960

suburbcity (to))from(

⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0

City population in 2001 = people who remained from 2000 + peopleCity population in 2001 = people who remained from 2000 + peoplewho moved in from the suburbs = ( 096 who moved in from the suburbs = ( 096 timestimes 58 ) + (001 58 ) + (001 timestimes 142 ) = 142 ) = 571 million571 millionSuburban population in 2001 = people who moved in from the city Suburban population in 2001 = people who moved in from the city + people who stayed from 2000 = ( 004 + people who stayed from 2000 = ( 004 timestimes 58 ) + (099 58 ) + (099 timestimes 142 ) = 142 ) = 1429 million1429 millionCan arrive at these numbers using matrix multiplicationCan arrive at these numbers using matrix multiplication

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01



Assume that the population flow represented by the matrix P is unchanged over the years The population distribution X2 after 2 years is given by X2 = P X1 = P2 X0 In general

Pn is called the n-step transition matrix The (ij)th element of Pngives the probability of going from state j to state i in n stepsFor example

in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=

The probability of living in the city in The probability of living in the city in 2000 and being in the suburbs 4 years 2000 and being in the suburbs 4 years later is 015later is 015

Markov ChainsMarkov Chains

The probabilities in the above model depend only on the The probabilities in the above model depend only on the current state of a person whether the person is living in the current state of a person whether the person is living in the city or in suburbiacity or in suburbiaThis type of model in which the probability of going from This type of model in which the probability of going from one state to another depends only on the current state one state to another depends only on the current state rather than on a more complete historical description is rather than on a more complete historical description is called a called a Markov chainMarkov chainMarkov chains are useful tools for scientists in many Markov chains are useful tools for scientists in many fields such as geneticsfields such as genetics


43

GeneticsGenetics 1414

Genetics is the branch of biology that deals with Genetics is the branch of biology that deals with heredity heredity （遺傳）（遺傳） It is the study of units called It is the study of units called genes which determine the characteristics living genes which determine the characteristics living things inherit from their parents The inheritance things inherit from their parents The inheritance （遺傳）（遺傳）of such traitsof such traits（特徵）（特徵）as sex height eye as sex height eye color and hair color of human beings and such color and hair color of human beings and such traits as petaltraits as petal（花瓣）（花瓣） color and leaf shape of color and leaf shape of plants are governed by genesplants are governed by genesIn medicine agricultureIn medicine agriculturehelliphellip


The traits of guinea pigsThe traits of guinea pigs（天竺鼠）（天竺鼠）are long hair and short hairare long hair and short hairThe length of hair is governed by a pair of gene which we denoteThe length of hair is governed by a pair of gene which we denote A A and aand aA guinea pig may have any one of the combination of AA A guinea pig may have any one of the combination of AA AaAa or or aaaa Each of these classes is called a genotypeEach of these classes is called a genotype（基因型）（基因型）The AA type guinea pig is indistinguishable in appearance from tThe AA type guinea pig is indistinguishable in appearance from the he AaAa type type ndashndash both have long hair both have long hair ndashndash whereas the whereas the aaaa type has short hairtype has short hairThe A gene is said to dominate the a geneThe A gene is said to dominate the a geneAn animal is called dominant An animal is called dominant （顯性）（顯性）if it has AA genes hybridif it has AA genes hybrid（混種）（混種） with with AaAa genes and recessive genes and recessive （隱性）（隱性）with with aaaa genesgenes


When two guinea pigs are crossed the offspring inherits one genWhen two guinea pigs are crossed the offspring inherits one gene e from each parent in random mannerfrom each parent in random mannerGiven the genotype of the parents we can determine the Given the genotype of the parents we can determine the probabilities of the genotype of the offspringprobabilities of the genotype of the offspring

AA+AA AA+AA probabilities of AA probabilities of AA AaAa and and aaaa are 10and 0are 10and 0Aa+AAAa+AA probabilities of AA probabilities of AA AaAa and and aaaa are are frac12frac12 frac12frac12 and 0 and 0aa+AAaa+AA probabilities of AA probabilities of AA AaAa and and aaaa are 0 1 and 0are 0 1 and 0

This series of experiments is a Markov chain having transition This series of experiments is a Markov chain having transition matrixmatrix

aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=


Assume an initial population of guinea pigs made up of an Assume an initial population of guinea pigs made up of an equal number of each genotypeequal number of each genotype

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0

The fractions of following generations are

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa

3-4 Systems of Linear Equations ndash Computational Aspects

DEFINITION 36 Equivalent

Two systems of linear equations are called equivalent if they have the same solution

set

兩個線性方程組稱為等價意指兩者具有相同的解集合

Elementary Transformations Transformation called elementary transformations can be used to change a system of

linear equations into another system of linear equations that have the same solution

These transformations are used to solve system of linear equations by eliminating

variables

The equivalent transformations used in terms of matrix are called ELEMENTARY

ROW OPERATION


44

Equivalent System Systems of equations that are related through elementary transformations are called

equivalent systems

Matrices that are related through elementary row operations are called row equivalent

matrices

The symbol asymp is used to indicate equivalence in both cases

Theorem 313 Let Ax = b be a system of m linear equations in n unknowns and let C be an invertible

mtimesm matrix Then the system (CA)x = Cb is equivalent to Ax = b

設 Ax = b 為含 n 個未知數m 個線性方程式的方程組若 C 為可逆的mtimesm

矩陣則(CA)x = Cb 等價於 Ax = b

【Proof】

Let K be the solution set for Ax = b and Krsquo the solution set for (CA)x = Cb If wisinK

then Aw = b So (CA)w = Cb and hence wisinKrsquo Thus KsubeKrsquo

Conversely if wisinKrsquo then (CA)w = Cb Hence Aw = C-1(CA)w = C-1(Cb) = b so

wisinK Thus KrsquosubeK and therefore K=Krsquo

設 K 為 Ax = b 的解集合且 Krsquo為(CA)x = Cb 的解集合

若 wisinK則 Aw = b (CA)w = Cb故 wisinKrsquo於是 KsubeKrsquo

反之若 wisinKrsquo則(CA)w = Cb Aw = C-1(CA)w = C-1(Cb) = b故 wisinK於是

KrsquosubeK因此K=Krsquo

Corollary Let Ax = b be a system of m linear equations in n unknowns If (Arsquo brsquo) is obtained ∣

from from (A b) by a finite number of elementary row operations then the system Arsquox = ∣

brsquo is equivalent to the original system

設 Ax = b 為含 n 個未知數m 個線性方程式的方程組若(Arsquo brsquo) ∣ 是由(A b) ∣

經過有限次數的基本列運算而得則 Arsquox = brsquo 等價於 Ax = b即兩者具有相同的解

集合

【Proof】

Suppose that (Arsquo brsquo) is obtained from (A b) by elementary row operations These ∣ ∣


45

may be executed by multiplying (A b) by elementary mtimesm matrices E∣ 1 E2hellipEp Let

C=EphellipE2E1 then (Arsquo brsquo) = C(A b) = (CA Cb)∣ ∣ ∣

Since each Ei is invertible so is C Now Arsquo = CA and brsquo = Cb Thus by Theorem 313

the system Arsquox = brsquo is equivalent to the system Ax = b

設(Arsquo brsquo)∣ 是由(A b)∣ 經過基本列運算取得即以基本 mtimesm 矩陣 E1 E2hellipEp

乘上 (A b)∣ (Arsquo brsquo) = C(A b) = (CA Cb∣ ∣ ∣ )其中C=EphellipE2E1

因 Ei 可逆故 C 亦為可逆 Arsquo = CA 且 brsquo = Cb

依據 Theorem 313Arsquox = brsquo 等價於 Ax = b

Theorem 313 Let Ax = b be a system of m linear equations in n unknowns and let C be an

invertible mtimesm matrix Then the system (CA)x = Cb is equivalent to Ax = b 設 Ax = b

為含 n 個未知數m 個線性方程式的方程組若 C 為可逆的mtimesm 矩陣則(CA)x

= Cb 等價於 Ax = b

We now describe a method for solving any stystem of linear equations Consider for

example the system of linear equations

探討解線性方程組的方法hellip

2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++

First we form the augmented matrix ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）

Interchanging the first and third rows ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121

（第 13 列互調）

Adding -1 times the first row to the second row and then add -3 times the first row to

the third row ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121

（第 1 列乘上 -1 加到第 2 列再以 -3 乘上

第 1 列加到第 3 列）


46

Multiplying the second row by -1 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）

Adding four times the second row to the third row ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2

列加到第 3 列）

Multiplying the third row by -13 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）

Adding the third row to the first and second rows to obtain zeros in row 1 column 4

and row 2 column 4 ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121

（將第 3 列加到第 1 列及第 2 列）

Adding -2 times the second row to the first row in order to make the first row second

column entry become zero ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101

（將第 2 列乘以 -2 再加至第 1 列）

We have now obtained the desired reduction of the augmented matrix

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101

是化簡後的增廣矩陣

This matrix corresponds to the system of linear equations

對應的線性方程組為

2x3x1xx

4

2

31

===+

By Corollary to Theorem 313 this system is equivalent to the original system But this

system is easily solved Obviously x2 = 2 and x4 = 3 Moreover x1 and x3 can be any values


47

provided their sum is 1 Letting x3 = t we then have x1 = 1 ndash t Thus an arbitray solution to the

original system has the form

依據 Corollary to Theorem 313 化簡後的方程組與原方程組等值此方程組的解為

x2 = 3x4 = 3x1 + x3 = 1於是原方程組的任意解可以寫成

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101

is a basis for the homogeneous system of equations

corresponding to the given system

其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101

是齊次方程組解集合的一組基底

Gauss-Jordan Elimination

Let A be an n times n matrix

1 Adjoin the identity n times n matrix In to A to form the matrix [A In]

2 Compute the reduced echelon form of [A In]

(1) If the reduced echelon form is of the type [In B] then B is the inverse of A

(2) If the reduced echelon form is not of the type [In B] in that the first n times n

submatrix is not In then A has no inverse

An n times n matrix A is invertible if and only if its reduced echelon form is In

DEFINITION 37 Reduced Row Echelon Form

A matrix is said to be in reduced row echelon form if the following three conditions

are satisfied


48

一個矩陣可以被稱為簡約列梯形式必須具備有下列三條件

(a) Any row containing a nonzero entry preceeds any row in which all the entries are

zero (if any)

含有非零元素的任一列排在全含零元素的任一列前面（儘可能）

(b) The first nonzero entry in each row in the only nonzero entry in its column

各列中第一個非零元素為其所在行中唯一的非零元素

(c) The first nonzero entry in each row is 1 and it occurs in a column to thr right of the

first nonzero entry in the preceeding row

各列中第一個非零元素為 1 且它出現在前一列中第一個非零元素的右邊某一

行中

Reduced Echelon Form A matrix is in reduced echelon form if

1 Any rows consisting entirely of zeros are grouped at the bottom of the matrix

2 The first nonzero element of each other row is 1 This element is called a leading 1

3 The leading 1 of each row after the first is positioned to the right of the leading 1 of

the previous row

4 All other elements in a column that contains a leading 1 are zero

Theorem 314 Gaussian Elimination Transformation Gaussian elimination transform any matrix into its reduced row echelon form

利用 Gaussian elimination 將矩陣轉換成簡約列梯形式

We now describe a method for solving a system in which the augmented matrix is in

reduced row echelon form To illustrate this procedure we consider the system

14x8x3x2x2x28x5x2xxx6x3xxxx17x9x4xx3x2

54321

54321

54321

54321

=minus+++=minus+++=minus+++=minus+++


49

for which the augmented matrix is

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）

Applying Gaussian elimination to the augmented matrix of the system produces the

following sequence of matrices

利用 Gaussian elimination method 處理增廣矩陣

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201

The system of linear equations corresponding to thus last matrix is

對應最後矩陣的線性方程組 2x2x

1xxx3x2x2x

54

532

531

=minus=+minus=minus+

Assign x3 = t1 x5 = t2

2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=

Thus an arbitray solution s is of the form

於是任意解 s 具有下列形式

s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112

is a basis for the solution set of the corresponding

homogeneous system of equation and

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013

is a particular solution to the original system

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013

是原方程組的特別解

EXAMPLE 1

The matrix ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101010011

is not in reduced row echelon form

因為第一行含有第一個非零元素在第一列但第一行尚含有其他非零元素


⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010


因為第一列第一個非零元素不在第一列第一個非零元素的右邊

The matrix ⎟⎟⎠

⎞⎜⎜⎝

⎛010002


因為第一列的第一個非零元素不是 1


51

Reduced Echelon FormReduced Echelon Form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801

Not in Reduced Echelon Form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071

In Reduced Echelon Form

Use GaussUse Gauss--Jordan elimination to Jordan elimination to find reduced echelon formfind reduced echelon form 1313

Step 1 Interchange rows if necessary to bring a nonzero Step 1 Interchange rows if necessary to bring a nonzero element to the top of the first nonzero column This nonzero element to the top of the first nonzero column This nonzero element is called a pivotelement is called a pivot

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1


Step 2 Create a 1 in the pivot location by multiplying the Step 2 Create a 1 in the pivot location by multiplying the pivot row by 1pivotpivot row by 1pivot

Step 3 Create zero elsewhere in the pivot column by adding Step 3 Create zero elsewhere in the pivot column by adding suitable multiples of the pivot row to all other rows of the suitable multiples of the pivot row to all other rows of the matrixmatrix

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1

The matrix is the reduced echelon form of the given matrixThe matrix is the reduced echelon form of the given matrix


Step 4 Cover the pivot row and all rows above it Repeat Steps Step 4 Cover the pivot row and all rows above it Repeat Steps 1 1 and 2 for the remaining and 2 for the remaining submatrixsubmatrix Repeat Step 3 for the whole Repeat Step 3 for the whole matrix Continue until the reduced echelon form is reachedmatrix Continue until the reduced echelon form is reached

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111

Theorem 315 Let Ax = b be a system of r nonzero equations is in n unknown Suppose that rank(A)

= rank (A b)∣ and that (A b)∣ is in reduced row echelon form Then

設 Ax = b 為含有 n 個未知數r 個非零方程式的方程組假設 rank(A) = rank

(A b)∣ 且 (A b)∣ 為簡約列梯形式於是

(a) rank(A) = r

(b) If the general solution obtained by the procedure above is of the form s = s0 + t1u1

+ t2u2 +hellip+ tn-run-r then u1 u2hellip un-r is a basis for the solution set of the

corresponding homogeneous system and s0 is a solution to the original system 由

前述方法所得的 General solution 具有下列形式

s = s0 + t1u1 + t2u2 +hellip+ tn-run-r

其中u1 u2hellip un-r為齊次方程組 Ax = 0 的解集合的一組基底且 s0 為原方

程組的一個解）

【Proof】

Since (A b)∣ is in reduced row echelon form (A b)∣ must have r nonzero rows

Clearly these rows are linearly independent by the definition of the reduced row echelon


52

form and so rank (A b)∣ = r Thus rank(A) = r

Let K be the solution set for Ax = b and let KH be the solution set for Ax = 0 Setting

t1 = t2 =hellip= tn-r = 0 we see that s = s0isinK But by Theorem 39 K = s +KH Hence

KH = - s0+K = span(u1 u2hellip un-r )

Because rank(A) = r we have dim(KH) = n-r Thus since dim(KH) = n-r and KH is

generated by a set u1 u2hellip un-r containing at most n-r vectors that is a basis for KH

由於(A b)∣ 具有簡約列梯形式因此(A b)∣ 必定具有 r 個非零列由簡約列梯形

式的定義得知 r 個非零列為線性獨立故 rank (A b)∣ = rrank(A) = r

令 K 為 Ax = b 的解集合 KH 為 Ax = 0 的解集合

假設 t1 = t2 =hellip= tn-r = 0（u1 u2hellip un-r非 KH的基底）則 s = s0isinK

依據 Theorem 39K = s +KH KH = - s0+K = span(u1 u2hellip un-r )

因 rank(A) = r所以 dim(KH) = n - r 且 KH 是由u1 u2hellip un-r 所生成

故原假設「t1 = t2 =hellip= tn-r = 0」不成立

因此 KH含有最多 n - r 個向量且這些向量為 KH的基底

Theorem 39 Let K be the solution set of a system of linear equations Ax = b and let KH be

the solution set of the corresponding homogeneous system Ax = 0 Then for any solution s

to Ax = b K =s + KH = s+k kisinKH 設 K 為線性方程組 Ax = b 的解集合且 KH

為對應的齊次方程組的解集合則對 Ax = b 的任一個解 s 而言K =s + KH = s+k

kisinKH

ExampleExampleSolve the system of equationsSolve the system of equations

7537429333

321

321

321

=minusminus=+minus=+minus

xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32

31+minus=+minus=rArr=+

=+rArr xxxx

xxxx

The general solution to the system isThe general solution to the system is

parameter) a (callednumber real is where 1243

3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample

Solve the system of equationsSolve the system of equations446942

1423258101242

4321

4321

4321

=minus+minusminus=+minus+minus

=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminusminusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321

=+minus+minusminus=++minus+=++minus+

xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasympharr 210000

642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusminusminusasympminus+ 210000

321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr

2xs and r somefor 1x sx rx

3sr2x

5

432

1

=minus===

++minus=

ExampleExample

Solve the system of Solve the system of equationsequations 122

32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asympminusminusminus

minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131

The system has no solutionThe system has no solution

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1

Curve FittingCurve Fitting

A set of data points A set of data points is given and it is is given and it is necessary to find a polynomial whose graph passes through the necessary to find a polynomial whose graph passes through the points points The points are often measurements in an experiment The points are often measurements in an experiment The xThe x--coordinates are called coordinates are called base pointsbase pointsIt can be shown that if the base points are all distinct then aIt can be shown that if the base points are all distinct then a unique unique polynomial of degree n polynomial of degree n ndashndash 1 (or less)1 (or less)

can be fitted to thecan be fitted to the pointspoints

) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K

The coefficients can be found by substituting the points The coefficients can be found by substituting the points into the polynomial equation and then solving a system into the polynomial equation and then solving a system of linear equationsof linear equations

ExampleExample

Determine the equation of the polynomial of degree 2 whose graphpasses through the points (1 6) (2 3) (3 2)

Let the polynomial be and substituting x = 1 y = 6 x = 2 y = 3 x = 3 y = 2 we have

Solving this system for a0 a1 and a2 using Gauss-Jordan elimination

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321

We get a0 = 11 a1 = ndash 6 a2 = 1The parabola that passes through these points is y = 11 ndash 6x + x2

ExampleExample Traffic FlowTraffic Flow

Network analysis plays an important role in electrical Network analysis plays an important role in electrical engineeringengineeringThe concepts and tools of network analysis have been The concepts and tools of network analysis have been found to be useful in many other fields such as found to be useful in many other fields such as information theory and the study of transportation systemsinformation theory and the study of transportation systemsThe analysis of traffic flow through a road network during The analysis of traffic flow through a road network during peak period illustrates how systems of linear equations peak period illustrates how systems of linear equations with with manymany solutions can arise in practicesolutions can arise in practice

Traffic FlowTraffic Flow

Consider the typical road network The streets are all oneConsider the typical road network The streets are all one--way with the arrows indicating the direction of traffic flowway with the arrows indicating the direction of traffic flowThe flow of traffic in and out of the network is measured The flow of traffic in and out of the network is measured in vehicles per hour (in vehicles per hour (vphvph))How to construct a mathematical model to analyze this How to construct a mathematical model to analyze this network network

Traffic Traffic FlowFlowTRAFFIC LAWTRAFFIC LAWhelliphellipAll traffic entering an intersection must leave that All traffic entering an intersection must leave that intersectionintersection


54

Junction AJunction BJunction CJunction DJunction EJunction F 600

800 200 500 100 600

54

643

73

72

621

51

=+=++minus=+minus=minus=+minus=+

xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L

Traffic FlowTraffic FlowJunction AJunction BJunction CJunction DJunction EJunction F 600

800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

Perform road work on x7 Because x3 ge 0 the minimum flow 200 in the branch x7 can be obtained by making x3 = 0 (closing DE to traffic)


We get

765

76473

72761

xx x600xxx200x x500x x600xxx

minus=++minus=minus=+=+minusminus=


2

EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A 經過四種不同型態的運算helliphellip

TYPE 1 Interchanging the second row of A with the first row The result matrix is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus=

210443213112

B

TYPE 2 Multiplying the second column of A by 3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431324361

C

TYPE 3 Adding 4 times the third row of A to the first row

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

21043112

127217M

TYPE 3 Adding -4 times the third row of M to the first row of M A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

Notice that if a matrix Q can be obtained from matrix P by means of an elementary row

operation the P can be obtained from Q by an elementary row operation of the same type

注意若矩陣 Q 經過基本列運算後得到矩陣 P則矩陣 P 亦可由同一 TYPE 的

基本列運算得到矩陣 Q

DEFINITION 32 Elementary matrix

An ntimesn elementary matrix is a matrix obtained by performing an elementary operation

on In The elementart matrix is said to be of type 1 2 or 3 according to whether the

elementary operation performed on In is type 1 2 or 3 operation respectively

「基本矩陣」係指在 In 上實施某一基本運算後所得的矩陣分別在 In 上實施

TYPE 123 基本運算後所得基本矩陣稱為 TYPE 123


3


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100001010


互調取得




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus

100010201

















B = EA [= AE]


4







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100001010

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus==

210443213112

EAB






= AE 取得

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000010000300001

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

C = AE = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

210431324361




【Proof】



5










⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

3I⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

010100001

1ER2 harr R3

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100050001

2E5R25R2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100012001

3ER2+ 2R1R2+ 2R1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

ihgfedcba

A

AEAfedihgcba

1

010100001

=sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡R2 harr R3

AEAihgfed

cba

2

100050001

555 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡5R2

AEAihg

cfbeadcba

3

100012001

222 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+++

R2+ 2R1



1221EI

RR +asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010021

1E

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

I

IE 2R21R1 minus


22R21REI

minusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minus=

100010021

E 1



( )( )

1

n

1

3

1

2

1

111

11

1n

1

n

1

3

1

2

1

1

1

n

1

3

1

2

1

11

EEEEABBAEE

EEEAEEEEEA


minusminus


=rArr==

=








6












DEFINITION 29








DEFINITION 211





sum=

=n

1iiiuax



7

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=β

n

2

1

a

aa


DEFINITION 212




sum==

m



β= [T]A





sum==

m

1iiijj wa)vT(




提示 sum==

m


DEFINITION 217







8











that sum=

=m








then





【Proof】




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




3


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100001010


互調取得




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus

100010201

















B = EA [= AE]


4







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100001010

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus==

210443213112

EAB






= AE 取得

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000010000300001

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

C = AE = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

210431324361




【Proof】



5










⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

3I⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

010100001

1ER2 harr R3

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100050001

2E5R25R2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100012001

3ER2+ 2R1R2+ 2R1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

ihgfedcba

A

AEAfedihgcba

1

010100001

=sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡R2 harr R3

AEAihgfed

cba

2

100050001

555 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡5R2

AEAihg

cfbeadcba

3

100012001

222 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+++

R2+ 2R1



1221EI

RR +asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010021

1E

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

I

IE 2R21R1 minus


22R21REI

minusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minus=

100010021

E 1



( )( )

1

n

1

3

1

2

1

111

11

1n

1

n

1

3

1

2

1

1

1

n

1

3

1

2

1

11

EEEEABBAEE

EEEAEEEEEA


minusminus


=rArr==

=








6












DEFINITION 29








DEFINITION 211





sum=

=n

1iiiuax



7

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=β

n

2

1

a

aa


DEFINITION 212




sum==

m



β= [T]A





sum==

m

1iiijj wa)vT(




提示 sum==

m


DEFINITION 217







8











that sum=

=m








then





【Proof】




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




4







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100001010

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ minus==

210443213112

EAB






= AE 取得

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

1000010000300001

E ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

210431124321

A

C = AE = ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

210431324361




【Proof】



5










⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

3I⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

010100001

1ER2 harr R3

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100050001

2E5R25R2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100012001

3ER2+ 2R1R2+ 2R1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

ihgfedcba

A

AEAfedihgcba

1

010100001

=sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡R2 harr R3

AEAihgfed

cba

2

100050001

555 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡5R2

AEAihg

cfbeadcba

3

100012001

222 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+++

R2+ 2R1



1221EI

RR +asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010021

1E

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

I

IE 2R21R1 minus


22R21REI

minusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minus=

100010021

E 1



( )( )

1

n

1

3

1

2

1

111

11

1n

1

n

1

3

1

2

1

1

1

n

1

3

1

2

1

11

EEEEABBAEE

EEEAEEEEEA


minusminus


=rArr==

=








6












DEFINITION 29








DEFINITION 211





sum=

=n

1iiiuax



7

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=β

n

2

1

a

aa


DEFINITION 212




sum==

m



β= [T]A





sum==

m

1iiijj wa)vT(




提示 sum==

m


DEFINITION 217







8











that sum=

=m








then





【Proof】




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




5










⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

3I⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

010100001

1ER2 harr R3

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100050001

2E5R25R2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100012001

3ER2+ 2R1R2+ 2R1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

ihgfedcba

A

AEAfedihgcba

1

010100001

=sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡R2 harr R3

AEAihgfed

cba

2

100050001

555 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡5R2

AEAihg

cfbeadcba

3

100012001

222 =sdot⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+++

R2+ 2R1



1221EI

RR +asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010021

1E

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

I

IE 2R21R1 minus


22R21REI

minusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minus=

100010021

E 1



( )( )

1

n

1

3

1

2

1

111

11

1n

1

n

1

3

1

2

1

1

1

n

1

3

1

2

1

11

EEEEABBAEE

EEEAEEEEEA


minusminus


=rArr==

=








6












DEFINITION 29








DEFINITION 211





sum=

=n

1iiiuax



7

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=β

n

2

1

a

aa


DEFINITION 212




sum==

m



β= [T]A





sum==

m

1iiijj wa)vT(




提示 sum==

m


DEFINITION 217







8











that sum=

=m








then





【Proof】




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




6












DEFINITION 29








DEFINITION 211





sum=

=n

1iiiuax



7

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=β

n

2

1

a

aa


DEFINITION 212




sum==

m



β= [T]A





sum==

m

1iiijj wa)vT(




提示 sum==

m


DEFINITION 217







8











that sum=

=m








then





【Proof】




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




7

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=β

n

2

1

a

aa


DEFINITION 212




sum==

m



β= [T]A





sum==

m

1iiijj wa)vT(




提示 sum==

m


DEFINITION 217







8











that sum=

=m








then





【Proof】




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




8











that sum=

=m








then





【Proof】




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




9




Corollary






column



【Proof】










依據 Theorem 22






10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




10

span(a1 a2 an)






（1 j p≦ ≦ ）則

(a) uj = A vj


EXAMPLE 1

Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101110101

A





⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎭

⎪⎬

⎫

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111

010

101

span = 2

A 的第 1 行a1=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

101第 2 行a2=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

010第 3 行a3=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111






EXAMPLE 2 Let


11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




11

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

211301121

A


the result is

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220121



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

110220001





D = ⎟⎟⎠

⎞⎜⎜⎝

⎛

32

1r

OOOI


Dij = 0 otherwise



⎞⎜⎜⎝

⎛

32

1r

OOOI



矩陣 D

【Proof】







12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




12


Suppose that m = 1









gt1)













matrix


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0B

0001






13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




13









Dlsquo= ⎟⎟⎠

⎞⎜⎜⎝

⎛ minus

65

41r

OOOI



Let

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

0D

0001

D











各處皆為 0


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

192362100280844422420

A


14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




14




D

00000001000001000001

192362100280844422420

A =

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

==

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=




⎞⎜⎜⎝

⎛

32

1r

OOOI





⎞⎜⎜⎝

⎛

32

1r

OOOI


【Proof】















15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




15









⎞⎜⎜⎝

⎛

32

1r

OOOI







rows






Rank

【Proof】









16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




16


Thus






設 r = rank(A)






則










【Proof】





A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1


17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




17







於是A = B-1InC-1 = B-1C-1即 A = E1-1E2

-1hellipEp-1Gq

-1Gq-1-1hellipG1

-1










【Proof】







β]U[








18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




18

βα

γβ



γβ

γα = [T][U][UT]





EXAMPLE 4

(a) Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minus

=11111121


(b) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

003011011131


(c) Let ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus=

011111121321












19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




19








Thus )AI()IAAA(CA 1


n1

n11ppminusminus

minus ==





(InA-1)











20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




20







(InB)則 B=A-1








are zeros



EXAMPLE 5


⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420


compute its inverse




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

100010001

133242420




⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

100001010

133420242


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

10000100

133420121 2

1


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


230420121

23

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

minusminus 100000

230210121

23

21

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

10001

400210301

23

2321

21


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minusminus

41

83

8321

21

0001

100210301


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minus

41

83

83

21

43

41

43

85

81

100010001

目標達成


⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A


22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




22

EXAMPLE 6


⎠

⎞

⎜⎜⎜

⎝

⎛minus=451112


compute its inverse



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

100010001

451112




3

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

100010001

451112

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


101012001

330330

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


113012001

000330

121













23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




23


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=β

100210211





inverse

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=minus

β

100210

011)([T] 1



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

minus=++ β

minus

2

21

10

2

1

02

2101

aa2aaa

aaa

100210

011)]xaxa(a[T






γβγ = [u][T][T(u)]






24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




24



EXAMPLE

⎥⎦

⎤⎢⎣

⎡=

4321

A and ⎥⎦

⎤⎢⎣

⎡minus

minus=

21

23

12B

221

23

221

23

I1001

432112

BA

I100112

4321

AB

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡minus

minus=

=⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡minus

minus⎥⎦

⎤⎢⎣

⎡=







InB = C B = C







[ ] [ ][ ] [ ]n21n21

n21n21

CCCAXAXXACCCXXXALL

LL

=rArr=




Thus when A-1 exits

[ ] [ ] [ ] 1n



[ ][ ]n21n

n21


LL

L

asympasymp


25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




25


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=100531010532001211

]IA[ 3

⎥⎥⎦

⎤

⎢⎢⎣

⎡minusminusminus

minusminus

+minus+

asymp

101320012110001211

1RR1R3

2)(R2

⎥⎥⎦

⎤

⎢⎢⎣

⎡minus


101320012110001211

R2)1(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=531532211



⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus++asymp

123100012110013101

R2)2(R3R2R1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminusminus

minus++asymp

123100135010010001

R3)1(R2R3R1

123135110

A 1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

+minus+asymp

135000011210012301

3R2R3R2)1(R1


⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

412721511

A

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minus=

100412010721001511

][ 3IA

⎥⎥⎦

⎤

⎢⎢⎣

⎡


minus+minus+asymp

102630011210001511

R1)2(R31)R1(R2






ccA





⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

BGR

311523212321275596

114587299

QIY

YIQgtRGBYIQgtRGB



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡rArr

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

QIY

0751108116472721

6209561

BGR

QIY

311523212321275596

114587299

BGR 1




A A-1Message





26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




26



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

434110433


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusminus

2331372091431715418461146

222117196116169

27319225271527921112013272

434110433

EncodingEncoding



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus 27319225271527921112013272

2331372091431715418461146

222117196116169

334344101

DecodingDecoding



mnmn22m11m

2nn2222121

1nn1212111

bxaxaxa

bxaxaxabxaxaxa

=+++

=+++=+++

(S)



field F



The mtimesn matrix

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

mn2m1m

n22221

n11211

aaa

aaaaaa



⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

n

2

1

x

xx

x and

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

m

2

1

b

bb






27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




27


n

2

1

F

s

ss

s isin

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛








EXAMPLE 1


321

321

=+minus=++

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛minus 6

1

xxx

211132

3

2

1



⎠

⎞

⎜⎜⎜

⎝

⎛minus=

726

s and ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus=

34

8s

Consider 1xx0xx

21

21

=+=+

that is ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡10

xx

1111

2

1











28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




28








【Proof】


dimension theorem



















29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




29




【Proof】







EXAMPLE 2

Consider 0xxx

0xx2x

321

321

=minusminus=++

Let ⎟⎟⎠

⎞⎜⎜⎝

⎛minusminus

=111







⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32

1 is a basis for


⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t where tisinR

例如因⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32


具有如下形式

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

t3t2

t

32

1t


30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




30












【Proof】





So w - sisinKH











w = s + kisins +KH


31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




31

因此 Ksube s +KH


因 Aw = A(s + k) = As + Ak = b + 0 = b


因此s +KHsubeK

總結K=s +KH

EXAMPLE 3

Consider 4xxx7xx2x

321

321

minus=minusminus=++



⎠

⎞

⎜⎜⎜

⎝

⎛=

411



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K by Theorem 39

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

411


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus32



⎪⎬

⎫

⎪⎩

⎪⎨

⎧isin

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus+

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛= Rt

32

1t

411

K






32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




32





【Proof】





















1xx3x33x2x4x2

2x4x2

321

321

32

=++=++

=+




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




33

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

133242420

A⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus

41

83

83

21

43

41

43

85

81

1A

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

minus+minus

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


==⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛minus

814587

41

83

83

21

43

41

43

85

81

1

3

2

1

132

bAxxx



system Ax = b



【Proof】



column of A












b)

其條件相當於


34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




34








1xx0xx

21

21

=+=+

Since ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1111

A and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

111011

)bA(










2xx3xxx3xxx

31

321

321

=+=+minus=++



R(T)


T(s)=(3 3 2)


35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




35


31

321

321

=+=+minus=++

的解


⎠

⎞

⎜⎜⎜

⎝

⎛minus

101111111


⎠

⎞

⎜⎜⎜

⎝

⎛minus

233

101111111

的 Rank 為 3兩






增廣矩陣

Application




濟獎

CLOSED MODEL











36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




36




費比例如下


農夫 Farmer 040 020 020





his or her income










040 p1 + 020 p2 + 020 p3 = p1





37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




37

040 p1 + 020 p2 + 020 p3 = p1

010 p1 + 070 p2 + 020 p3 = p2

050 p1 + 010 p2 + 060 p3 = p3


其中⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

3

2

1

ppp

p 且⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

600100500200700100200200400








sumgt

jjkjk pAp


sumsumsumsum ==gt

jj

i jjij





⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p


38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




38


⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250

p



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

400350250


能生存下去

Theorem 312


⎞⎜⎜⎝

⎛=

EDCB





⎞⎜⎜⎝

⎛=

EDCB





⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

02502503502500650500750

A

OPEN MODEL






39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




39






demands Hence i

3


=

for j=1 2 3









i

3


=

for j=123







⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A





40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




40



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

300200300100400100300200300

A


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛


=minus=

minus

minus

700600900

100200300

700200300100600100300200300

d)AI(x

1

1

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

700600900



Theorem



【Proof】





gives



41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




41


2x5x3x3x5x3x21x2x x

321

321

321


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


231

xxx

531532211

3

2

1



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

231

531532211

xxx 1

3

2

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

121

231

123135110

xxx

3

2

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


=

minus

minus

123135110

531532211

A

1

1











nindustry of demand

2industry of demand

22

1industry of demand

11


ininii

i










in turn 321812

and 896

1612

9

5100

021

21

103

51

51

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= DA



42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




42




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus⎥⎥⎦

⎤

⎢⎢⎣

⎡=minus minus

minusminus

5400

021

21

103

51

54

5100

021

21

103

51

31

100010001

get We AI

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus minus

4500

038

35

85

32

35

1)( AI


4500

038

35

85

32

35

)(

401020883957522133

32816189121269

)(

1

1

DAI

DAIX

minus

minus


⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=minus=




40

15674

and 106629

209946

are 321812

and 896

1612

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡






1m

m2m2

m2

AI)AAAI(A)AAAI(I

)AAAI)(AI(


++++minusLL

L

NOTE 0A 1m rarr+




suburbcity

990040010960


⎥⎦

⎤⎢⎣

⎡=P

⎥⎦

⎤⎢⎣

⎡=

14258

X0


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

9142157

14258

990040010960

PXX 01





in

ni XPX =+

suburbcity

960150040850

P


4

⎥⎦

⎤⎢⎣

⎡=





43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




43









aaAaAA

00012100211

P

aaAaAA

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

313131

X0


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

01611615

X08187

X04143

X02121

X 4321

AAAAAaAaaaaa




set





variables


ROW OPERATION


44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




44


equivalent systems


matrices






【Proof】














集合

【Proof】



45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




45
















2xxx2x3xxx1x2x3x2x3

4321

321

4321

=minus++=++=minus++


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

231

112101112323

（增廣矩陣）


⎠

⎞

⎜⎜⎜

⎝

⎛

minus

minus

132

132301111121




⎠

⎞

⎜⎜⎜

⎝

⎛

minusminusminus

minus

512

104010101121




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




46


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

51

2

104010101121

（第 2 列乘上 -1）


⎠

⎞

⎜⎜⎜

⎝

⎛

minusminus

minusminusminus

91

2

300010101121

（4 乘上第 2



⎠

⎞

⎜⎜⎜

⎝

⎛minusminus

minus

31

2

100010101121

（第 3 列乘上 -13）



⎠

⎞

⎜⎜⎜

⎝

⎛

325

100000100121




⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101



⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

321

100000100101




2x3x1xx

4

2

31

===+




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




47





⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛ minus

0101

t

3021

3t2

t1

Observe that

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101



其中

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛minus

0101












are satisfied


48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




48



zero (if any)







行中





the previous row







54321

54321

54321

54321



49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




49


⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

（增廣矩陣）




⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


1486

17

83222521113111194132

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛


148

176

83222521119413231111

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

minusminus

minus

0213

0000021000

1011020201



1xxx3x2x2x

54

532

531



2t22x2x1tt1xxx

3t2t23x2x2x

254

21532

21531

+=+=+minus=+minus=

++minus=++minus=



s =

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

+

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

+

+minus++minus

=

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1201

2

t

00112

t

02013

t2t2

t1tt

3t2t2

xxxxx

21

2

2

1

21

21

5

4

3

2

1

where t1t2isinR


50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




50

Notice that

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112



⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛minus

1201

2

00112


⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

02013


EXAMPLE 1


⎠

⎞

⎜⎜⎜

⎝

⎛

101010011




⎠

⎞

⎜⎜⎜

⎝

⎛

110010012010




⎞⎜⎜⎝

⎛010002




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




51


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

910030107001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100001000041

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

000210801


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

310000004021

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

100000430003021

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000210030108071




⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

1211244129333

22200

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus

harrasymp

121124422200

129333

R2R1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+asymp

412002220043111

4)R1(R3

⎥⎥⎦

⎤

⎢⎢⎣

⎡

minusminus

minus

⎟⎠⎞

⎜⎝⎛asymp

12112442220043111

R13

1




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

minus++asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

⎟⎠⎞

⎜⎝⎛

asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

6100050100

170011

R3R22)R3(R1

610001110052011

R2)2(R3R2R1

412001110043111

R221

412002220043111





(a) rank(A) = r








【Proof】




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




52



















kisinKH


7537429333

321

321

321


xxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++asymp

000012104301

R2R3R2R1 12

431243

32

31

32


=+rArr xxxx

xxxx



3

2

1

rrxrxrx

=+minus=+minus=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminusminus

minus

minus+minus+asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


asymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


242012103111

1R)3(3R1R)2(2R

715374123111

715374129333

R131

ExampleExample


1423258101242

4321

4321

4321


=minus+minus

xxxxxxxxxxxx

⎪⎪

⎩

⎪⎪

⎨

⎧

===

minus=

rArr==minus=minus

rArr

1x6xrx

2r2x

1x6x

2x2x

4

3

2

1

4

3

21

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminusasymp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡


minusminus

+

minus+

+

minus+

minus+

+

110006010020021

110005110011021

14430051100

295621

2R31

144300153300295621

446942142321295621

44694214232158101242

R3R2

1)R3(R1

3R2R3

6)R2(R1

2)R1(R3

R1R2R1

21


53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




53


432636242232

54321

54321

54321


xxxxxxxxxxxxxxx

⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

⎥⎥⎦

⎤

⎢⎢⎣

⎡


+

minus+

642000210000213121

431121636242213121

R1R3

2)R1(R2

⎥⎥⎦

⎤

⎢⎢⎣


642000213121

R3R2 ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡ minusasymp

210000321000213121

2R21

⎥⎥⎦

⎤

⎢⎢⎣


321000750121

3)R2(R1 ⎥⎥⎦

⎤

⎢⎢⎣

⎡minus

minusasymp

minus+

+

210000101000300121

2)R3(R2

5R3R1

2x1x

3xx2x

5

4

321

=minus=

++minus=rArr


3sr2x

5

432

1

=minus===

++minus=

ExampleExample


32452232

32

321

321

321

=+minus=minus+

=+minus=+minus

xxxxxxxxxxx

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

harr1220210063303211

R3R2

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminusminus

minus

1220210021103211

R231

⎥⎥

⎦

⎤

⎢⎢

⎣


minus+

+

5400210021101101

2)R2(R4

R2R1 ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

minus+

+

minus+

13000210040103001

4)R3R(R4

R3R2

1)R3(R1

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

minusminus

1000210040103001

R4131


⎥⎥

⎦

⎤

⎢⎢

⎣

⎡asymp

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

minusminusminus

minus


minus+

minus+

1220633021003211

1220312145223211

1)R1(R3

2)R1(R2

0x1+0x2+0x3=1




) ( ) ( ) ( 2211 nn yxyxyx K

11

2210

minusminus

minusminus ++++= n

nn

n xaxaxaay K


ExampleExample




⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minus

minus+minus+asymp

220033109201

1)R2(R31)R2(R1

2210 xaxaay ++=

2a9a3a3a4a2a6aaa

210

210

210

=++=++=++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minusminus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

minus+minus+asymp

482033106111

293134216111

1)R1(R31)R1(R2

( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡minus

minusasymp

minus++asymp

11006010

11001

110033109201

3)R3(R22R3R1

R321








54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761




54


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

asympasymp

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

minusminus


minusminus

minusminus

0000000001110000

6001101000200100010050010000106001100001

600001100080001011002001000100500100001010001000116000010001

L


800 200 500 100 600

54

643

73

72

621

51


xxxxx

xxxx

xxxxx



We get

765

76473

72761



chapter 3 elementary matrix operations and …linear algebra- chapter 3 2008/09/26 3 for example,...

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