matrix – basic definitions

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Matrix – Basic Definitions

Chapter 3 Systems of Differential Equations


Matrix – Properties

Matrices A, B and C with elements aij, bij and cij, respectively.

1. EqualityFor A and B each be m by n arrays

Matrix A = Matrix B if and only if aij = bij for all values of i and j.

2. Addition

A + B = C if and only if aij + bij = cij for all values of i and j.

For A , B and C each be m by n arrays

3. CommutativeA + B = B + A

4. Associative(A + B) + C = A + (B + C)

If B = O (the null matrix), for all A : A + O = O + A = A

0..00

........

0..00

0..00

O

5. Multiplication (by a Scalar)αA = (α A)

in which the elements of αA are α aij



Matrix Multiplication, Inner Product

CAB if and only if k

kjikijbac

Matrix multiplication

* In general, matrix multiplication is not commutative !

BAAB commutator bracket symbol 0BAAB]B,A[

But if A and B are each diagonal BAAB

* associative )BC(AC)AB(

* distributive ACAB)CB(A

The product theorem

For two n × n matrices A and B BAAB




2221

1211

3231

2221

1211

232221

131211

cc

cc

bb

bb

bb

aaa

aaa

22322322221221

21312321221121

12321322121211

11311321121111

)()()(

)()()(

)()()(

)()()(

cbababa

cbababa

cbababa

cbababa

Successive multiplication of row i of A with column j of B – row by column multiplication

For example :[2 × 3] × [3 × 2] = [2 × 2]




4518

4716

3811

6*05*91*02*9

6*25*71*22*7

6*35*41*32*4

61

52

09

27

34

AB

For example :

[3 × 2] × [2 × 2] = [3 × 2]



Unit Matrix, Null Matrix

0..00

........

0..00

0..00

O

The unit matrix 1 has elements δij, Kronecker delta, and the property that 1A = A1 = A for all A

1..00

........

0..10

0..01

1

The null matrix O has all elements being zero !

Exercise 3.2.6(a) : if AB = 0, at least one of the matrices must have a zero determinant.

If A is an n × n matrix with determinant 0, then it has a unique inverse A-1 so that AA -1 = A -1 A = 1.



Direct product --- The direct tensor or Kronecker product

If A is an m × m matrix and B an n × n matrix

The direct product CBA

C is an mn × mn matrix with elements

klijBAC k)1i(n l)1j(n with

For instance, if A and B are both 2 × 2 matrices

)

cccc

cccc

cccc

cccc

()

babababa

babababa

babababa

babababa

()BaBa

BaBa(BAC

44434241

34333231

24232221

14131211

2222212222212121

1222112212211121

2212211222112111

1212111212111111

2221

1211

The direct product is associative but not commutative !



Diagonal Matrices

If a 3 × 3 square matrix A is diagonal

33

22

11

a00

0a0

00a

A

In any square matrix the sum of the diagonal elements is called the trace.

i

iia)A(trace

1. The trace is a linear operation : )B(trace)A(trace)BA(trace 2. The trace of a product of two matrices A and B is independent of the order of multiplication : (even though AB BA)

)BA(trace)BA(abba)AB()AB(tracej

jjj i

ijjii j

jiiji

ii

0)BA(trace)AB(trace])B,A([trace 3. The trace is invariant under cyclic permutation of the matrices in a product. )CAB(trace)BCA(trace)ABC(trace



Matrix Inversion

Matrix A An operator that linearly transforms the coordinate axes

Matrix A-1 An operator that linearly restore the original coordinate axes

1AAAA 11

A

Ca)A( ji)1(

ijij

1 The elements Where Cji is the jith cofactor of A.

For example :

43

21A The cofactor matrix C

12

34C

10A A

C)A( ji

ij

1

13

24

10

1A 1

and 0A



Matrix Inversion

For example :

121

012

113

A |A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2

),1(

),1(

),1(

31

21

11

c

c

cThe elements of the cofactor matrix are

),2(

),4(

),2(

32

22

12

c

c

c

),5(

),7(

),3(

33

23

13

c

c

c

5.25.35.1

0.10.20.1

5.05.05.0

573

242

111

2

1

A

CA

T

1



Special matrices

A matrix is called symmetric if:

AT = A

A skew-symmetric (antisymmetric) matrix is one for which:

AT = -A

An orthogonal matrix is one whose transpose is also its inverse:

AT = A-1

Any matrix ]A~

A[2

1]A

~A[

2

1A

symmetric antisymmetric



Inverse Matrix, A-1

xAx ' '1 xAx

The reverse of the rotationxAAxAx 1'1 1AA 1

Transpose Matrix, A~

Defining a new matrix such that A~

ijjiaa~

jki

ikijaa jk

iikji

aa~ 1AA~

11 AAA~

A 1AA~

holds only for orthogonal matrices !



Eigenvectors and Eigenvalues

vvA A is a matrix, v is an eigenvector of the matrix and λ the corresponding eigenvalue.

0 vvA

0

3

2

1

3

2

1

333231

232221

131211

v

v

v

v

v

v

aaa

aaa

aaa

0

3

2

1

333231

232221

131211

v

v

v

aaa

aaa

aaa

This only has none trivial solutions for det (A- λ I) = 0. This gives rise to the secular equation for the eigenvalues:

0

333231

232221

131211

aaa

aaa

aaa



Eigenvectors and Eigenvalues



Example 3.5.1 Eigenvalues and Eigenvectors of a real symmetric matrix

000

001

010

aaa

aaa

aaa

A

333231

232221

131211

The secular equation

0

00

01

01

λ = -1,0,1

0

z

y

x

00

01

01

λ = -1. x+y = 0, z = 0

)0,2

1,

2

1(r

1

Normalized

λ = 0 x = 0, y = 0

)1,0,0(r2

λ = 1 -x+y = 0, z = 0

Normalized

Normalized)0,2

1,

2

1(r

3



Example 3.5.2 Degenerate Eigenvalues

010

100

001

aaa

aaa

aaa

A

333231

232221

131211

The secular equation

0

10

10

001

λ = -1,1,1

0

z

y

x

10

10

001

λ = -1. 2x = 0, y+z = 0

)2

1,

2

1,0(r

1

Normalized

λ = 1 -y+z = 0 (r1 perpendicular to r2)

)2

1,

2

1,0(r

2

λ = 1

Normalized

Normalized)0,0,1(rrr213

(r3 must be perpendicular to r1 and may be made perpendicular to r2)




Conversion of an nth order differential equation to a system of n first-order differential equations

)y,,y,y,t(Fy )1n(')n(

Setting , , , ……yy1 '2 yy ''

3 yy )1n(n yy

2'1 yy

3'2 yy

4'3 yy

……

)y,,y,y,t(Fy n21'n

Ayy '

xAx

txey tt' Axexey



Example : Mass on a spring

0ym

ky

m

cy ''' 2

'1 yy 21

'2 y

m

cy

m

ky

2

1

'2

'1

y

y

m

c

m

k10

y

y0

m

k

m

c

m

c

m

k1

)IAdet( 2

5.01 5.11

assume 1m 2c 75.0k

0xx5.0 21 eigenvector

1

2c

x

x1

2

1

0xx5.1 21

5.1

1c

x

x2

2

1

eigenvector

t5.12

t5.01

2

1 e5.1

1ce

1

2c

y

y

t5.12

t5.011 ecec2y



Homogeneous systems with constant coefficients

Ayy ' in components212111

'1 yayay

222121'2 yayay

)t(y

)t(y)t(y

2

1y1y2-plane is called the phase plane

Critical point : the point P at which dy2/dy1 becomes undetermined is called

212111

222121'1

'2

1

2

1

2

yaya

yaya

y

y

dt/dy

dt/dy

dy

dy

P : (y1,y2) = (0,0)



Five Types of Critical points



Criteria for Types of Critical points

0Adet)aa(aa

aa)IAdet( 2211

2

2221

1211

2211 aap 21122211 aaaaAdetq q4p2

21212

212 )())((qp

P is the sum of the eigenvalues, q the product and the discriminant.



Stability Criteria for Critical points



Example : Mass on a spring

0ym

ky

m

cy ''' 2

'1 yy 21

'2 y

m

cy

m

ky

2

1

'2

'1

y

y

m

c

m

k10

y

y

p = -c/m , q = k/m and = (c/m)2-4k/m

No damping c = 0 : p = 0, q > 0 a center

Underdamping c2 < 4mk : p < 0, q > 0, < 0 a stable and attractive spiral point.

Critical damping c2 = 4mk : p < 0, q > 0, = 0 a stable and attractive node.

Overdamping c2 > 4mk : p < 0, q > 0, > 0 a stable and attractive node.



No basis of eigenvectors available. Degenerate node

Ayy ' If matrix A has a double eigenvalue

t)1( xey tt)2( uextey

tt)2(ttt)'2( AueAxteAyuextexey

since Axx

Auux xu)IA(

If matrix A has a triple eigenvalue

ttt2)3( veuteext2

1y uv)IA(



No basis of eigenvectors available. Degenerate node

2

1

'2

'1

y

y

21

14Ay

y

y0)3(96

21

14)IAdet( 22

1

1x

u

u

11

11u)I3A(

2

1

1

0

u

u

2

1

t32

t31

)2(2

)1(1 e)

1

0t

1

1(ce

1

1cycycy

matrix – basic definitions

Documents

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rotationtranspose matrix

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cofactor matrix arechapter

cofactor matrix candchapter

commutativea b