2- 1 chapter 2 matrices definition of a matrix. 2- 2 a system of 3 equations: represented by a...
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2- 1
Chapter 2 Matrices
• Definition of a matrix
3231
2221
1211
A columns) 2 rows, (3matrix 23 (a)
aa
aa
aa
rcrr
c
c
bbb
bbb
bbb
21
22221
11211
Bmatrix c r (b)
2- 2
6
4
24
-34
24
34
1
2
21
1
3
21
1
3
21
E
PLd
L
E
wLd
L
E
wLd
L
A system of 3 equations:
Represented by a matrix:
1
21
31
3
6
411
24
341
24
314
E
PL
L
E
wL
L
E
wL
L
2- 3
Types of Matrices• Square matrix: # of rows = # of columns
• upper triangular matrix strictly upper triangular matrix
333231
232221
131211
aaa
aaa
aaa
0000
000
00
0
55
4544
353433
25242322
1514131211
a
aa
aaa
aaaa
aaaaa
00000
0000
000
00
0
45
3534
252423
15141312
a
aa
aaa
aaaa
2- 4
• lower triangular matrix strictly lower triangular matrix
• diagonal matrix
0
00
000
0000
5554535251
44434241
333231
2221
11
aaaaa
aaaa
aaa
aa
a
0
00
000
0000
00000
54535251
434241
3231
21
aaaa
aaa
aa
a
000
0 00
0 00
0 00
n
1
2
1
2- 5
• banded matrix
a square matrix with elements of zero except for the principal d
iagonal and values in the positions adjacent to the diagonal.
• tridiagonal matrix
000
00
00
00
000
5554
454443
343332
232221
1211
aa
aaa
aaa
aaa
aa
2- 6
• unit matrix: 1 on the principal diagonal
• null matrix: All elements are zero.
10000
01000
00100
00010
00001
Ι
00000
00000
00000
00000
00000
O
2- 7
• symmetric matrix: a square matrix in which
• skew-symmetric matrix: a square matrix in which for all i a
nd j
jiij aa
1.000.640.27-
0.641.000.23-
0.27-0.23-1.00
jiij aa
2- 8
• transpose of matrix A: AT
• (AT) T = A
jiTij aa
5.63.84.61.77.7
55188531235
60132283195140
5.65560
3.8188132
4.653283
1.712195
7.735140
TAA
2- 9
Matrix Operations
• Matrix equality
• Matrix addition and subtractionC = A + B = B + A (commutative)
C = A - B
ijijij bac
ijijij bac
jiba ijij and allfor if BA
2- 10
• Example: Matrix addition and subtraction
11109
531
642
A
112
487
320
B
121111
9118
962
BAC
1097
156
322
BAD
2- 11
Matrix Multiplication
• One example
8143
55 35
4(9)8(3)3(7)4(3)8(2)3(5)
1(9)6(3)4(7))3(1)2(6)5(4
93
32
75
483
164
CST
2- 12
Rules of Matrix Multiplication
1. # of columns in A = # of rows in B
2. # of rows in C = # of rows in A
3. # of columns in C = # of columns in B
4.
BAC
m
kkjikij bac
1
2- 13
5. Matrix multiplication is not commutative
6. Matrix multiplication is associativeABBA
)())( CBACBA
2- 14
Example: Matrix Multiplication
11109
531
642
A
112
487
320
B
7810992
203131
284240
BAE
282114
1269258
433629
ABF
ABBA
2- 15
Matrix Multiplication by a Scalar
ijij sabs AB
10
11109
531
642
sA
11010090
503010
604020
AB s
An example:
2- 16
Matrix Inversion
where A-1 is the inverse of A, and I is the unit matrix
IAA 1
1 c
0 c
0 c
1 c
22221221
21221121
22121211
21121111
aca
aca
aca
aca
equations ussimultaneo following by the
determined be can inverse the,)( and 2 If2n
cn ij 1A
2- 17
Example: Matrix Inversion
10
01
75
32
2221
1211
cc
cc
75
32A
1 73
0 52
0 73
1 52
2221
2221
1211
1211
cc
cc
cc
cc
25
37
get we
1A
2- 18
Matrix Singularity
• If the inverse of a matrix A exists, then A is said to be nonsingular.
• If the inverse of a matrix A does not exist, then A is said to be singular.
• If matrix A is singular, then the linear system of simultaneous equations represented by A has no unique solution.
2- 19
There are an infinite number of solutions if 2a = b.
There is no feasible solution if 2a b.
Thus matrix A is singular.
bXX
aXX
21
21
64
32
64
32 Let
A
10
01
64
32for solution No
2221
1211
cc
cc
2- 20
• trace of a square matrix = sum of diagonal elements
• matrix augmentation: addition of a column or columns
to the initial matrix
n
iiiatr
1
)(A
100
010
001
432
141
132
432
141
132
aAA
2- 21
• matrix partition
2221
1211
AA
AAA
432
141
132
A
4 32
1
1
41
32
2221
1211
AA
A A
2- 22
Vectors• Column vector
• Row vector
• Vectors of two ordinates
132
2
1
2
2- 23
• orthogonal vectors
Two vectors are said to be orthogonal if their product is equal to zero.
If two vector are orthogonal, they are perpendicular to each other in the n-dimensional space.
01
32 example,For 3
2
2- 24
50
1
2lengthvector .n
ii )v(
• normalized vectors
A vector is normalized by dividing each element by its length.
A normalized vector has a length 1.
Two vectors that are both normalized and orthogonal to each other are said to be orthonormal vectors.
2- 25
Example: Vectors
5] 3- [21 V
1
1
1
2
V
6.16438)5()3((2) of length 2221 V
732.13)1()1((-1) of length 2222 V
2- 26
38
5
38
3
38
21nV
3
1
3
1
3
12nV
Normalized vectors:
l.orthonorma are and
.orthogonal are and ,0 Since
21
11
nn VV
VVVV 22
2- 27
Determinants• A determinant of a matrix A is denoted by |A|.
• The determinant of a 22 matrix:
• The determinant of a 33 matrix:
bcadc d
a b
a a
a a a
a a
a a a
a a
a a a
a a a
a a a
a a a
3231
222113
3331
232112
3332
232211
333231
232221
131211
2- 28
• The minor of aij, denoted by Aij, is the matrix after removing row i and column j.
• The determinant of an nn matrix:
• The general expression for the determinant of an nn matrix:
||a1)(||a||a||a|| 1n1n
131211 1n131211 AAAAA
||)1(||)1(||)1(||)1(|| 333
222
111
ininni
iii
iii
iii aaaa AAAAA
2- 29
Example: Matrix Determinant
• with the first row and their minors:
11109
531
642
A
|||||||| 131211 131211 AAAA aaa
0)]9(3)10(1[6)]9(5)11(1[4)]10(52[3(11)
109
316
119
514
1110
532
11109
531
642
||
A
2- 30
• with the second column and their minors:
• Since |A|=0, A is a singular matrix; that is the inverse of A doest not exist.
|||||||| 232212 232212 AAAA aaa
0]610[10]5422[3]4511[4
51
6210
119
623
119
514
11109
531
642
||
A
11109
531
642
A
2- 31
Properties of Determinants
1. If the values in any row (column) are proportional to the corresponding values in another row(column), the determinant equals zero
0|| where,
353
2142
121
AA
0|| where,
653
4142
221
AA
2- 32
2. If all the elements in any row(column) equal zero, the determinant equals zero.
3. If all the elements of any row(column) are multiplied by a constant c, the value of the determinant is multiplied by c.
14)]4(2)5(3[2|| where,54
)2(2)3(2
54
46
AA
2- 33
4. The value of the determinant is not changed by adding any row (column) multiplied by a constant c to another row (column).
5. If any two rows (columns) are interchanged, the sign of the determinant is changed.
7)]4(2)5(3|| where,54
23
AA
7)4(3)5(1|| where,54
3-1-
BB
-73(5)-2(4)45
32 and 72(4)-3(5)
54
23
2- 34
6. The determinant of a matrix equals that of its transpose; that is, |A| = |AT|.
7. If a matrix A is placed in diagonal form, then the product of the elements on the diagonal equals the determinant of A.
74(2)-3(5)52
43 and 72(4)-3(5)
54
23
7)3
7(3
3
70
03||
3
70
03
3
70
23
72(4)3(5)|A| with,54
23
AA
A
2- 35
8. If a matrix A has a zero determinant, then A is a singular matrix; that is, the inverse of A does not exist.
2- 36
Rank of A Matrix
• A matrix of r rows and c columns is said to be of order r by c. If it is a square matrix, r by r, then the matrix is of order r.
• The rank of a matrix equals the order of highest-order nonsingular submatrix.
2- 37
3 square submatrices:
Each of these has a determinant of 0, so the rank is less than 2. Thus the rank of R is 1.
Example 1: Rank of Matrix
842
421 matrix,order 32 R
84
42 ,
82
41 ,
42
21321
RRR
2- 38
Since |A|=0, the rank is not 3. The following submatrix has a nonzero determinant:
Thus, the rank of A is 2.
Example 2: Rank of Matrix
11109
531
642
A
2)1(4)3(231
42
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