2005/8 Matrices-1

Matrices

2005/8 Matrices-2

A Matrix over a Field F (R or C)

nm

mnmmm

n

n

n

ij M

aaaa

aaaa

aaaa

aaaa

aA

][

nm321

3333231

2232221

1131211

m rows

n columns

size: m×n

ij-entry: aij F (ij-component)

2005/8 Matrices-3

The i-row (vector)

The j-column (vector)

),,,( 21 iniii aaaA

mj

j

j

j

a

a

a

A2

1

(row matrix)

(column matrix)

Square matrix: m = n

2005/8 Matrices-4

Diagonal matrix

),,,( 21 nddddiagA nn

n

M

d

d

d

00

00

00

2

1

Tracennija A ][

nnaaaATr 2211)(

unit nn matrix = diag(1, 1, , 1) zero matrix = diag(0, 0, , 0)

2005/8 Matrices-5

Example:

654

321A

,3211 A 6542 A

,4

11

A ,

5

22

A

6

33A

2005/8 Matrices-6

nmijnmij bBaA ][ ,][

Equal

njmibaBA ijij 1 ,1 ifonly and if

Example:

dc

baBA

43

21

4,3,2,1 ifonly and if dcbaBA

2005/8 Matrices-7

Matrix addition

nmijnmij bBaA ][ ,][

nmijijnmijnmij babaBA ][][][

Example:

31

50

2110

3211

21

31

10

21

2

3

1

2

3

1

22

33

11

0

0

0

2005/8 Matrices-8

Matrix Subtraction

BABA )1(

Scalar Multiplication over a field F (R or C)

F cAA nm ,][

nmijcacA ][

2005/8 Matrices-9

Matrix Multiplication

pnijnmij bBaA ][ ,][

pmijpnijnmij cbaAB ][][][

equalSize of AB

njin

n

kjijikjikij babababac

1

2211

inijii

nnnjn

nj

nj

nnnn

inii

n

cccc

bbb

bbb

bbb

aaa

aaa

aaa

21

1

2221

1111

21

21

11211

2005/8 Matrices-10

05

24

31

A

14

23B

Example:

Sol:

)1)(0()2)(5()4)(0()3)(5(

)1)(2()2)(4()4)(2()3)(4(

)1)(3()2)(1()4)(3()3)(1(

AB

1015

64

19

2005/8 Matrices-11

The partitioned matrices

2221

1211

34333231

24232221

14131211

AA

AA

aaaa

aaaa

aaaa

A

submatrix

3

2

1

34333231

24232221

14131211

r

r

r

aaaa

aaaa

aaaa

A

4321

34333231

24232221

14131211

cccc

aaaa

aaaa

aaaa

A

2005/8 Matrices-12

Properties of Matrix Operations Three elementary matrix operations:

(1) addition

(2) scalar multiplication

(3) multiplication

zero matrix: nm0

identity matrix of order n: nI

2005/8 Matrices-13

The properties of addition and scalar multiplication

(1) A + B = B + A

(2) A + ( B + C ) = ( A + B ) + C

(3) ( cd ) A = c ( dA )

(4) 1A = A

(5) c( A+B ) = cA + cB

(6) ( c+d ) A =cA + dA

thenscalars, are , and ,,, If dcMCBA nm

2005/8 Matrices-14

The properties of zero matrix

Note：(1) 0m×n: the addition identity ( 加法單位矩陣 )

(2) A: the addition inverse ( 加法反元素 ) of matrix A

If AMmn, and c is a scalar, then

(1) A + 0mn = A

(2) A + (A) = 0mn

(3) cA = 0mn c = 0 or A = 0mn

2005/8 Matrices-15

The properties of matrix multiplication

(1) Ａ (BC) = (AB)C

(2) Ａ (B+C) = AB + AC

(3) (A+B)C = AC + BC

(4) c(AB) = (cA) B = A(cB)

The properties of identity matrix

AAI nmMA

n

)1(then

, If

AAI m )2(

2005/8 Matrices-16

The transpose ( 轉置 ) of a matrix

nm

mnmm

n

n

M

aaa

aaa

aaa

A

If

21

22221

11211

mn

mnnn

m

m

T M

aaa

aaa

aaa

A

then

21

22212

12111

2005/8 Matrices-17

Ex: Find the transpose of the following matrices.

8

2A

(b)

987

654

321

A

(c)

11

42

10

A

Sol: (a)

8

2A 82 TA

(b)

987

654

321

A

963

852

741TA

(c)

11

42

10

A

141

120TA

(a)

2005/8 Matrices-18

)4(

)3(

)2(

)1(

TTT

TT

TTT

TT

ABAB

AccA

BABA

AA

The properties of transpose matrices

2005/8 Matrices-19

The symmetric matrix ( 對稱矩陣 )If A = AT , then the square matrix A is called symmetric.

If AT = A , then the square matrix A is called skew-symmetric.

Example:

6

54

321

If

cb

aA is symmetric, then find the values of a, b, c.

Sol:

5 ,3 ,2 cba

The skew-symmetric matrix ( 反對稱矩陣 )

6

54

321

cb

aA

653

42

1

c

ba

AT

TAA

2005/8 Matrices-20

Ex:

0

30

210

If

cb

aA is skew-symmetric, find a, b, c.

Sol:

3 ,2 ,1 cba

Note: TAA is symmetric.

Pf:

symmetric. is

)()(T

TTTTTT

AA

AAAAAA

,

0

30

210

cb

aA

032

01

0

c

ba

AT

TAA

2005/8 Matrices-21

Real Numbers

ab = ba Multiplication commutative

Matrices

BAAB pnnm

undefined. is

defined. is then, If

BA

ABpm (1)

mm

mm

MBA

MABn pm

(3) then, If

nn

nm

MBA

MABnmpm

(2) then, , If

Three possibilities:

2005/8 Matrices-22

Ex: For given matrices A and B, show that AB BA.

12

31A and

20

12B

Sol:

44

52

20

12

12

31AB

BAAB

24

70

12

31

20

12BA

2005/8 Matrices-23

Real numbers

ac = bc, 0c

ba Cancellation laws

Matrices

0 CBCAC

(1) If C is invertible, then A = B

(2) If C is non-invertible, then . (cancellation law does not hold)

BA

2005/8 Matrices-24

Ex： For given matrices A, B and C, show that AC=BC.

21

21 ,

32

42 ,

10

31CBA

Sol:

21

42

21

21

10

31AC

BCAC , but .BA

21

42

21

21

32

42BC