2 - 1

Chapter 2AChapter 2A MatricesMatrices

2A.1 Definition, and Operations of Matrices: 1 Sums and Scalar Products; 2 Matrix Multiplication

2A.2 Properties of Matrix Operations; Mechanics of Matrix Multiplication

2A.3 The Inverse of a Matrix

2A.4 Elementary Matrices

2 - 2

2A.1 Operations with Matrices

Matrix:

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

[ ]

n

n

nij m n

m m m mn m n

a a a a

a a a a

a a a aA a M

a a a a

L

L

L

M M M M

L

(i, j)-th entry: ija

row: m

column: n

size: m×n

2 - 3

i-th row vector

1 2i i i inr a a a L

j-th column vector

1

2

j

j

j

mj

c

cc

c

M

row matrix

column matrix

Square matrix: m = n

2 - 4

Diagonal matrix:

1 2( , , , )nA diag d d d L

1

2

0 0

0 0

0 0

n n

n

d

dM

d

K

L

M M O M

L

Trace:

If [ ]ij n nA a

11 22Then ( ) nnTr A a a a L

2 - 5

Ex:

654

321A

2

1

r

r

321 ccc

,3211 r 6542 r

,4

11

c ,

5

22

c

6

33c

654

321A

2 - 6

Let [ ] , [ ]ij m n ij m nA a B b

Equal matrix:

if and only if 1 , 1ij ijA B a b i m j n

Ex 1: (Equal matrix)

dc

baBA

43

21

BA If

4 ,3 ,2 ,1 Then dcba

2 - 7

Matrix addition:

If [ ] , [ ]ij m n ij m nA a B b

Then [ ] [ ] [ ]ij m n ij m n ij ij m nA B a b a b

Scalar multiplication:

scalar : ,][ If caA nmij

nmijcacA ][Then

2 - 8

Matrix subtraction:

BABA )1(

Example: Matrix addition

31

50

2110

3211

21

31

10

21

2

3

1

2

3

1

22

33

11

0

0

0

2 - 9

Matrix multiplication:

The i, j entry of the matrix product is the dot product of row i of the left matrix with column j of the right one.

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Matrix multiplication:

Notes:Notes: (1) A+B = B+A, (2) BAAB

11 12 111 1 1

21 2 2

1 21 2

11 2

nj n

j n

i i ini i ij in

n nj nnn n nn

a a a b b b

b b ba a a

c c c c

b b ba a a

L L LM M M

M LL

L LM M M MM M M

L LL

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Matrix form of a system of linear equations:

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

L

L

M

L

= = =

A x b

equationslinear m

equationmatrix Single

A x b 1m n n 1m

11 12 1 1 1

21 22 2 2 2

1 2

n

n

m m mn n m

a a a x b

a a a x b

a a a x b

L

L

M M M M M M

L

2 - 12

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

2 - 13

11 12 1

21 22 21 2

1 2

n

nn

m m mn

a a a

a a aA c c c

a a a

L

LL

M M M M

L

1

2

n

x

xx

x

M

11 1 12 2 1

21 1 22 2 2

1 1 2 2 1

n n

n n

m m mn n m

a x a x a x

a x a x a xAx

a x a x a x

L

L

M

L

a linear combination of the column vectors of matrix A:

linear combination of column vectors of A

11 21 1

21 22 21 2

1 2

n

nn

m m mn

a a a

a a ax x x

a a a

LM M M

1c

=

2c

=

nc

=

2 - 14

Keywords in Section 2.1:

row vector: 列向量 column vector: 行向量 diagonal matrix: 對角矩陣 trace: 跡數 equality of matrices: 相等矩陣 matrix addition: 矩陣相加 scalar multiplication: 純量積 matrix multiplication: 矩陣相乘 partitioned matrix: 分割矩陣

2 - 15

2A.2 Properties of Matrix Operations

Three basic matrix operators:

(1) matrix addition

(2) scalar multiplication

(3) matrix multiplication

Zero matrix:nm0

Identity matrix of order n: nI

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Then (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

scalar:, ,,, If dcMCBA nm

Properties of matrix addition and scalar multiplication:

Commutative law for addition

Associative law for addition

Associative law for scalar multiplication

Unit element for scalar multiplication

Distributive law 1 for scalar multiplication

Distributive law 2 for scalar multiplication

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calar cMA nm s: , If

AA nm 0 (1)Then

nmA A 0)((2)

nmnm or A c cA 000)3(

Notes:

(1) 0m×n: the additive identity for the set of all m×n matrices

(2) –A: the additive inverse of A

Properties of zero matrices:

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Properties of the identity matrix:

AAI

MA

n

nm

(1)Then

If

AAIm (2)

Properties of matrix multiplication:

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11 12 1

21 22 2

1 2

If

n

nm n

m m mn

a a a

a a aA M

a a a

L

L

M M M M

L

11 21 1

12 22 2

1 2

Then

m

mTn m

n n mn

a a a

a a aA M

a a a

L

L

M M M M

L

Transpose of a matrix:

AA TT )( )1(TTT BABA )( )2(

)()( )3( TT AccA

)( )4( TTT ABAB

Properties of transposes:

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A square matrix A is symmetric if A = AT

Ex:

6

54

321

If

cb

aA is symmetric, find a, b, c?

A square matrix A is skew-symmetric if AT = –A

Skew-symmetric matrix:

Sol:

6

54

321

cb

aA

653

42

1

c

ba

AT

5 ,3 ,2 cba

TAA

Symmetric matrix:

2 - 21

0

30

210

If

cb

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

Note:Note: TAA is symmetric

Pf:

symmetric is

)()(T

TTTTTT

AA

AAAAAA

Sol:

030210

cbaA

032

01

0

c

ba

AT

TAA 3 ,2 ,1 cba

Ex:

2 - 22

ab = ba (Commutative law for multiplication)

undefined. is defined is then , If BAABpm ， (1)

mmmm MBAMABnpm ， (3) then , If

nnmm MBAMABnmpm ， (2) then , , If (Sizes are not the same)

(Sizes are the same, but matrices are not equal)

Real number:

Matrix:

A B B A m n n p

Three situations of the non-commutativity may occur :

Åã¥Ü®à ±.scf

2 - 23

12

31A

20

12B

Sol:

44

52

20

12

12

31AB

24

70

12

31

20

12BA

Example (Case 3):

Show that AB and BA are not equal for the matrices.

and

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(Cancellation is not valid)

0 , cbcac

b a (Cancellation law)

Matrix:

0 CBCAC

(1) If C is invertible (i.e., C-1 exists), then A = B

Real number:

BA then ,invertiblenot is C If (2)

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21

21 ,

32

42 ,

10

31CBA

Sol:

21

42

21

21

10

31AC

So BCAC

But BA

21

42

21

21

32

42BC

Example: An example in which cancellation is not valid

Show that AC=BCC is noninvertible,

(i.e., row 1 and row 2

are not independent)

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Keywords in Section 2.2:

zero matrix: 零矩陣 identity matrix: 單位矩陣 transpose matrix: 轉置矩陣 symmetric matrix: 對稱矩陣 skew-symmetric matrix: 反對稱矩陣

2 - 27

2A.3 The Inverse of a Matrix

nnMA

,such that matrix a exists thereIf nnn IBAABMB

Note:Note:

A matrix that does not have an inverse is called

noninvertible (or singular).

Consider

Then (1) A is invertible (or nonsingular)

(2) B is the inverse of A

Inverse matrix:

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If B and C are both inverses of the matrix A, then B = C.

Pf:

CB

CIB

CBCA

CIABC

IAB

)(

)(

Consequently, the inverse of a matrix is unique.the inverse of a matrix is unique. Notes:Notes:(1) The inverse of A is denoted by 1A

IAAAA 11 )2(

TheoremTheorem:: The inverse of a matrix is unique

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1nEliminatioJordan -Gauss || AIIA

Example: Find the inverse of the matrix

31

41A

Sol:IAX

10

01

31

41

2221

1211

xx

xx

10

01

33

44

22122111

22122111

xxxx

xxxx

Find the inverse of a matrix AA by Gauss-Jordan Elimination:

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1 2 2 1; 41 4 1 1 0 3(1)

1 3 0 0 1 1

M M

M M

1 2 2 1; 41 4 0 1 0 4(2)

1 3 1 0 1 1

M M

M M

1 ,3 2111 xx

1 ,4 2212 xx

11

431AX

Thus

(2) 1304

(1) 0314

2212

2212

2111

2111

xxxx

xxxx

2 - 31

1 2 2 1

Gauss JordanElimination

; 4

1

1 4 1 0 1 0 3 4

1 3 0 1 0 1 1 1

A I I A

M M

M M

If A can’t be row reduced to I, then A is singular.

Note:

2 - 32

326101011

A

1 2

1 1 0 1 0 0

0 1 1 1 1 0

6 2 3 0 0 1

M

M

M

Sol:

1 1 0 1 0 0

1 0 1 0 1 0

6 2 3 0 0 1

A I

M

M M

M

1 36

1 1 0 1 0 0

0 1 1 1 1 0

0 4 3 6 0 1

M

M

M

3

1 1 0 1 0 0

0 1 1 1 1 0

0 0 1 2 4 1

M

M

M

2 34

1 1 0 1 0 0

0 1 1 1 1 0

0 0 1 2 4 1

M

M

M

Example: Find the inverse of the following matrix

2 - 33

3 2

1 1 0 1 0 0

0 1 0 3 3 1

0 0 1 2 4 1

M

M

M

2 1

1 0 0 2 3 1

0 1 0 3 3 1

0 0 1 1 4 1

M

M

M

So the matrix A is invertible, and its inverse is

1 [ ]I A M

You can check: IAAAA 11

1

2 3 1

3 3 1

1 4 1

A

2 - 34

0(1) A I

factors

(2) ( 0)k

k

A AA A k L144424443

(3) , : integersr s r sA A A r s

( )r s rsA A

1 1

2 2

0 0 0 0

0 0 0 0(4)

0 0 0 0

k

kk

kn n

d d

d dD D

d d

L LL L

M M O M M M ML L

Power of a square matrixsquare matrix:

2 - 35

If A is an invertible matrix, k is a positive integer, and c is a scalar,

then 1 1 1(1) is invertible and ( )A A A

1 1* 1 1 1 1

factors

(2) is invertible and

( ) ( )

k

k k k k

k

A

A A A A A A A L1444442444443

1 11(3) c is invertible and ( ) , 0A cA A c

c

1 1(4) is invertible and ( ) ( )T T TA A A

TheoremTheorem ：： Properties of inverse matrices

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TheoremTheorem:: The inverse of a product

If A and B are invertible matrices of size n, then AB is invertible and

111)( ABAB

111)( So

unique. is inverse its then ,invertible is If ABAB

AB

Pf:

IBBIBBBIBBAABABAB

IAAAAIAIAABBAABAB

1111111

1111111

)()()())((

)()()())((

Note:Note:

1 1 1 1 11 2 3 3 2 1n nA A A A A A A A

L L

2 - 37

Theorem:Theorem: Cancellation properties

If C is an invertible matrix, then the following properties

hold:

(1) If AC=BC, then A=B (Right cancellation property)

(2) If CA=CB, then A=B (Left cancellation property)

Pf:

BA

BIAI

CCBCCA

CBCCAC

BCAC

)()(

)()(11

11 exists) C so ,invertible is (C 1-

Note:If C is not invertible, then cancellation is not valid.

2 - 38

TheoremTheorem:: Systems of linear equations with unique solutions

If A is an invertible matrix, then the system of linear equations

Ax = b has a unique solution given by1x A b

Pf:

( A is nonsingular)

bAx

bAIx

bAAxA

bAx

1

1

11

This solution is unique.

.equation of solutions twowere and If 21 bAxxx

21then AxbAx 21 xx (Left cancellation property)

2 - 39

Keywords in Section 2.3:

inverse matrix: 反矩陣 invertible: 可逆 nonsingular: 非奇異 singular: 奇異 power: 冪次

2 - 40

Note:Note:

Only do a single elementary row operation.

2A.4 Elementary Matrices

Row elementary matrix:

An nn matrix is called an elementary matrixelementary matrix if it can be obtained

from the identity matrix I by a single elementary row operation.

2 - 41

100030001

)(a

010001)(b

000010001

)(c

010100001

)(d

1201)(e

100020001

)( f

Yes 2 3(3 ( ))I square)(not Not)constan nonzero aby bemust

tionmultiplica (Row No

Y 2 3 3es ([ ]( ))I Y 1 2 2es ([2 ]( ))I

)operations row

elementary two(Use No

Ex: (Elementary matrices and nonelementary matrices)

2 - 42

Notes:Notes:

(1) ( )ij ijA P A

(2) k ( ) ( )i iA M k A

i(3) [k ]( ) ( )j ijA C k A

( )

( )

I E

A EA

Theorem:Theorem: Representing elementary row operations

Let E be the elementary matrix obtained by performing an

elementary row operation on Im. If that same elementary row

operation is performed on an mn matrix A, then the

resulting

matrix is given by the product EA.

2 - 43

0262

2031

5310

A

Sol:

1 12 3

0 1 0

( ) 1 0 0

0 0 1

E I

2 1 3 3

1 0 0

[ 2 ]( ) 0 1 0

2 0 1

E I

3 3 3

12

1 0 01

[ ]( ) 0 1 02

0 0

E I

Example: Using elementary matrices

Find a sequence of elementary matrices that can be used to write

the matrix A in row-echelon form.

2 - 44

1 12 1

0 1 0 0 1 3 5 1 3 0 2

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

0 0 1 2 6 2 0 2 6 2 0

A A E A

2 1 3 1 2 1

1 0 0 1 3 0 2 1 3 0 2

[ 2 ]( ) 0 1 0 0 1 3 5 0 1 3 5

2 0 1 2 6 2 0 0 0 2 4

A A E A

3 3 2 3 2

1 0 0 1 3 0 2 1 3 0 21

[ ]( ) 0 1 0 0 1 3 5 0 1 3 52

1 0 0 2 4 0 0 1 20 0

2

A A E A

3 2 1 B E E E A 3 1 3 12

1or [ ][ 2 ] ( )

2B A

B

row-echelon form

2 - 45

Matrix B is row-equivalent to A if there exists a finite number

of elementary matrices such that

1 2 1k kB E E E E A L

Row-equivalent:Row-equivalent:

2 - 46

TheoremTheorem:: Elementary matrices are invertible

If E is an elementary matrix, then exists and

is an elementary matrix.

Notes:Notes:1(1) (P )ij ijP

1 1(2) [ ( )] ( )i iM k M

k

1(3) [C ( )] ( )ij ijk C k

1E

2 - 47

1 12 12

0 1 0

( ) 1 0 0

0 0 1

E P

1 112 1

0 1 0

( ) 1 0 0

0 0 1

P E

2 1 3 13

1 0 0

( 2 ) 0 1 0 ( 2)

2 0 1

E C

1 113 2

1 0 0

[ ( 2)] 0 1 0

2 0 1

C E

3 3 3

12

1 0 01 1

( ) 0 1 0 ( )2 2

0 0

E M

1 13 3

1 0 01

[ ( )] 0 1 02

0 0 2

M E

12P

13(2)C

3(2)M

Ex:

Elementary Matrix Inverse Matrix

2 - 48

Pf:Pf: (1) Assume that A is the product of elementary matrices.

(a) Every elementary matrix is invertible.

(b) The product of invertible matrices is invertible.

Thus A is invertible.

(2) If A is invertible, has only the trivial solution.0xA 0 0A I M M

3 2 1 kE E E E A I L1 1 1 1

1 2 3 kA E E E E L

Thus A can be written as the product of elementary matrices.

Theorem:Theorem: A property of invertible matrices

A square matrix A is invertible if and only if it can be written as

the product of elementary matrices.

2 - 49

83

21A

Sol:

1 1 2

22 1

3

1[ 2 ]2

1 2 1 2 1 2

3 8 3 8 0 2

1 2 1 0

0 1 0 1

A

I

21 2 12 1

1Therefore ( 2) ( ) ( 3) ( 1)

2C M C M A I

Ex:

Find a sequence of elementary matrices whose product is

2 - 50

1 1 1 11 12 2 21

1Thus [ ( 1)] [ ( 3)] [ ( )] [ ( 2)]

2A M C M C

1 12 2 21 ( 1) (3) (2) (2)M C M C

10

21

20

01

13

01

10

01

Note:Note:If A is invertible

13 2 1[ ] [ ]kE E E E A I I A L M M

3 2 1Then kE E E E A IL

13 2 1kA E E E E L

2 - 51

If A is an nn matrix, then the following statements are equivalent.

(1) A is invertible.

(2) Ax = b has a unique solution for every n1 column matrix b.

(3) Ax = 0 has only the trivial solution.

(4) A is row-equivalent to In .

(5) A can be written as the product of elementary matrices.

Thm:Thm: Equivalent conditions

2 - 52

LUAL is a lower triangular matrix

U is an upper triangular matrix

A nn matrix A can be written as the product of a lowerlower

triangular matrixtriangular matrix LL and an upper triangular matrixupper triangular matrix UU, such as

Note:Note:

If a square matrix A can be row reduced to an upper triangular

matrix U using only the row operation of adding a multiple of one only the row operation of adding a multiple of one

rowrow to another (pivot)to another (pivot), then it is easy to find an LU-factorization of A.

2 1

1 1 11 2

k

k

E E E A U

A E E E U

A LU

L

L

LLUU-factorization:-factorization:

2 - 53

01

21 )( Aa

2102310031

)( Ab

1 211 2 1 2

1 0 0 2A U

Sol: (a)

12( 1)C A U

112 12[ ( 1)] (1)A C U C U LU

112 12

1 0[ ( 1)] (1)

1 1L C C

Ex: Ex: LULU-factorization-factorization

2 - 54

(b)

1 3 2 32 4

1 3 0 1 3 0 1 3 0

0 1 3 0 1 3 0 1 3

2 10 2 0 4 2 0 0 14

A U

23 13(4) ( 2)C C A U

1 113 23[ ( 2)] [ (4)]A C C U LU

1 113 23 13 23[ ( 2)] [ (4)] (2) ( 4)

1 0 0 1 0 0 1 0 0

0 1 0 0 1 0 0 1 0

2 0 1 0 4 1 2 4 1

L C C C C

2 - 55

If thenA LU LUx b

Let theny Ux Ly b

Two steps:Two steps:

(1) Write y = Ux and solve Ly = b for y

(2) Solve Ux = y for x

bAx

Solving Ax=bAx=b with an LU-factorization of A

2 - 56

Sol:

LUA

1400

310

031

142

010

001

2102

310

031

2021021353

321

32

21

xxxxx

xx

bLyUxy solve and，Let )1(

2015

142010001

3

2

1

yyy

14)1(4)5(220

422015

213

2

1

yyy

yy

Ex: Solving a linear system using LU-factorization

2 - 57

yUx system following theSolve )2(

1415

1400

310031

3

2

1

xxx

1)2(3535

2)1)(3(131

1

21

32

3

xx

xx

x

Thus, the solution is

1

2

1

x

So

2 - 58

Keywords in Section 2A.4:

row elementary matrix: 列基本矩陣 row equivalent: 列等價 lower triangular matrix: 下三角矩陣 upper triangular matrix: 上三角矩陣 LU-factorization: LU 分解