chap. 2 matrices 2.1 operations with matrices 2.2 properties of matrix operations 2.3 the inverse of...

Chap. 2Chap. 2MatricesMatrices

2.1 Operations with Matrices

2.2 Properties of Matrix Operations

2.3 The Inverse of a Matrix

2.4 Elementary Matrices

2.5 Applications of Matrix Operations

Ming-Feng Yeh Chapter 2 2-2

Matrix representations: An uppercase case: A, B, C, … A representative element enclosed in brackets: [aij], [bij]

A rectangular array of numbers:

Vector (column/row matrix): boldface lowercasea1, a2, …, an

2.1 2.1 Operations with MatricesOperations with Matrices

mnmm

n

n

aaa

aaa

aaa

21

22221

11211


DefinitionsDefinitions Equality of Matrices

Two matrices A = [aij] and B = [bij] are equal if they have the same size (mn) and aij = bij for 1 i m and 1 j n.

Matrix AdditionIf A = [aij] and B = [bij] are matrices of size mn, then their sum is the mn matrix given by A+B = [aij + bij].The sum of two matrices of different sizes is undefined.

Scalar Multiplication If A = [aij] is an mn matrix and c is a scalar, then the scalar multiplication of A by c is the mn matrix given by cA = [caij]

Section 2-1


Example 1Example 1Consider the four matrices

Matrices A and B are not equal because they are of different sizes.

Similarly, B and C are not equal. Matrices A and D are equal if and only if (iff) x = 3Remark: “p if and only if q” means that p implies q and

q implies p.

4

21,31,

3

1,

43

21

xDCBA

Section 2-1


Subtraction of MatricesSubtraction of Matrices If A and B are of the same size, AB represents the sum of

A and (B). That is, AB = A+(1)B = [aij bij].

cA dB = [caij dbij].

Example 3:

407

6410

1261

223313)1(23

3)1(3)4(031)3(3

043023213

231

341

002

212

103

421

33 BA

231

341

002

and

212

103

421

BA

Section 2-1


Matrix MultiplicationMatrix Multiplication If A = [aij] is an mn matrix and B = [bij] is an np matrix,

then the product AB is an mp matrix AB = [cij], where

njinjiji

n

kkjikij babababac

2211

1

pmpnnm

ABBA

mpmjmm

ipijii

pj

pj

npnjnn

pj

pj

mnmm

inii

n

n

cccc

cccc

cccc

cccc

bbbb

bbbb

bbbb

aaa

aaa

aaa

aaa

21

21

222221

111211

21

222221

111211

21

21

22221

11211

Section 2-1


Example 4Example 4Find the product AB, where and

05

24

31

A

14

23B

3231

2221

1211

14

23

05

24

31

cc

cc

cc

9)4)(3()3)(1(11 c 1)1)(3()2)(1(12 c

10)1)(0()2)(5(15)4)(0()3)(5(

6)1)(2()2)(4(4)4)(2()3)(4(

3231

2221

cc

cc

1015

64

19

23 22 23

Section 2-1


Example 5Example 5

111

001

242

212

301)(a

11

21

11

21)(c

1

1

2

321)(d

321

1

1

2

)(e

32663

175

2210

01

111

33321

321

642

BAAB

Matrix multiplication is not, in general, commutative.

Section 2-1


Systems of Linear EquationsSystems of Linear Equations Matrix Equation: Ax = b

A: coefficient matrix; x and b: column matrix (vector)

Example 6: Solve the matrix equation Ax = 0, where

3333232131

2323222121

1313212111

3

2

1

3

2

1

333231

232221

131211

bxaxaxa

bxaxaxa

bxaxaxa

b

b

b

x

x

x

aaa

aaa

aaa

0

0,,

232

121

3

2

1

0x

x

x

x

A

010

001

74

71

.,

7

4

1

3

2

1

Rtt

x

x

x

x

Section 2-1


Diagonal Matrix & Trace (p. 58)Diagonal Matrix & Trace (p. 58)

A square matrix

is called a diagonal matrix if all entries that not on the main diagonal are zero.

The trace of an nn matrix A is the sum of the main diagonal entries. That is,

nna

a

a

A

00

00

00

22

11

nnaaaATr 2211)(

Section 2-1


2.2 Properties of Matrix Operations2.2 Properties of Matrix Operations

Theorem 2.1Properties of Matrix Addition and Scalar Multiplication

If A, B, and C are mn matrices and c and d are scalars, then the following properties are true.1. A+B = B+A Commutative property of addition

2. A+(B+C) = (A+B)+C Associative property of addition

3. (cd)A = c(dA) Associative property of multiplication

4. 1A = A Multiplication identity

5. c(A+B) = cA + cB Distributive property

6. (c+d)A = cA + dA Distributive property

交換律結合律結合律乘法單位元素左分配律右分配律


Proof of Theorem 2.1Proof of Theorem 2.1 The proofs follow directly from the definitions of matrix

addition and scalar multiplication, and the corresponding properties of real numbers.

Let A = [aij] and B = [bij]

1. Use the commutative properties of addition of real numbers to writeA+B = [aij+bij] = [bij+aij] = B+A

5. Use the distributive properties (for real number) of multiplication over addition to write c(A+B) = [c(aij+bij)] = [caij+cbij] = cA+cB

Section 2-2


Zero Matrix & Additive IdentityZero Matrix & Additive Identity If A is an mn matrix and Omn is the mn matrix

consisting entirely of zeros, then A + Omn = A.

The matrix Omn is called a zero matrix, and it serves as the additive identity for the set of all mn matrices.

Theorem 2.2: Properties of Zero MatrixIf A is an mn matrix and c is a scalar, then the following properties are true.1. A + Omn = A.2. A + (A) = Omn. A is the additive inverse of A. 3. If cA = Omn, then c = 0 or A = Omn.

Section 2-2


Matrix EquationMatrix Equation Real Numbers m n Matrices

Ex. 2: Solve for X in the equation 3X+A = B, where

abx

abx

abaax

bax

0

)()(

ABX

ABOX

ABAAX

BAX

)()(

12

43and

30

21BA

32

32

34 2

22

64

3

1

30

21

12

43

3

1)(

3

1ABX

Section 2-2


Theorem 2.3Theorem 2.3 Properties of Matrix MultiplicationIf A, B, and C are matrices (with sizes such that the given

matrix products are defined) and c is a scalar, then the following properties are true.1. A(BC) = (AB)C Associative property2. A(B+C) = AB + AC Distribution property3. (A+B)C = AC + BC Distribution property4. c(AB) = (cA)B = A(cB)

Proof of Property 2: A: mn matrix, B: np matrix, C: np matrix. The entry in the ith row and jth column of A(B+C) is

The entry in the ith row and jth column of AB +AC is)()()( 222111 njnjinjjijji cbacbacba

)()( 22112211 njinjijinjinjiji cacacabababa equal

Section 2-2


NoncommutativityNoncommutativity A commutative property for matrix multiplication was

NOT listed in Theorem 2.3. If A is of size 23 and B is of size 33,

then the product AB is defined, but the product BA is not. Example 4: Show that AB and BA are not equal for the

matrices and

12

31A

20

12B

44

52

20

12

12

31AB

24

70

12

31

20

12BA

BAAB

Section 2-2


Cancellation PropertyCancellation Property It does NOT have a general cancellation property for

matrix multiplication. If AC = BC, it is NOT necessary true that A = B. Example 5: Show that AC = BC.

21

21,

32

42,

10

31CBA

21

42

21

21

10

31

21

42

32

42

10

31

AC

AB

BABCAC but,

Section 2-2


Identity Matrix & Theorem 4Identity Matrix & Theorem 4 A square matrix that has 1’s on the main diagonal

and 0’s elsewhere.

The identity matrix of order n:

Theorem 2.4: Properties of the Identity Matrix If A is a matrix of size mn, then the following properties are true.1. AIn = A. 2. ImA = A.

If A is a square matrix of order n, then AIn = InA = A.

1000

0100

0010

0001

nI

Section 2-2


Repeated MultiplicationRepeated Multiplication Repeated multiplication of a square matrix:

For a positive integer k, Ak is

A0 = In, where A is a square matrix of order n. Example 3: Find A3 for the matrix .

AAAAk

k factors

kjkj AAA jkkj AA )(

j and k are nonnegative integer.

03

12A

63

14

03

12

36

21

03

12

03

12

03

123A

Section 2-2


Theorem 2.5Theorem 2.5 Number of Solutions of a System of Linear Equations

For a system of linear equations in n variables, precisely one of the following is true.

1. The system has exactly one solution.

2. The system has an infinite number of solutions.

3. The system has no solution.

Section 2-2


The Transpose of a MatrixThe Transpose of a Matrix The transpose of a matrix is formed by

writing its columns as rows.

A matrix A is symmetric if A = AT. aij = aji, i j. a symmetric matrix must be square.

nmmnmmm

n

n

n

aaaa

aaaa

aaaa

aaaa

A

321

3333231

2232221

1131211

mnmnnnn

m

m

m

T

aaaa

aaaa

aaaa

aaaa

A

321

3332313

2322212

1312111

Section 2-2


Theorem 2.6Theorem 2.6Properties of Transpose If A and B are matrices (with sizes such that the given

matrix products are defined) and c is a scalar, then the following properties are true.1. Transpose of a transpose

2. Transpose of a sum

3. Transpose of a scalar multiplication

4. Transpose of a product

For any matrix A, the matrix is symmetric.

AA TT )(TTT BABA )(

)()( TT AccA TTT ABAB )(

TTTT CBACBA )( TTTT ABCABC )(TAA

Section 2-2

TTTTTT AAAAAApf )()(:


Example 9Example 9 Show that are equal.

Sol:

TTT ABAB )(

120

301

212

A

03

12

13

B

211

162)(

21

16

12

03

12

13

120

301

212TABAB

211

162

132

201

012

011

323TT AB

TTT ABAB )(

Section 2-2


Example 10Example 10 For the matrix

find the product and show that it is symmetric.

Sol:

12

20

31

A

TAA

325

246

5610

123

201

12

20

31TAA jiij aa

TTT AAAA )( TAASince , is symmetric.

Section 2-2


2.3 The Inverse of a Matrix2.3 The Inverse of a Matrix Definition of an Inverse of a Matrix

An nn matrix A is invertible (or nonsingular) if there exists an nn matrix B such that AB = BA = In

In is the identity matrix of order n. The matrix B is called the (multiplicative) inverse of A. A matrix that does NOT have an inverse is called

noninvertible (or singular). Nonsquare matrices do NOT have inverse. Theorem 2.7: Uniqueness of an Inverse Matrix

If A is an invertible matrix, then its inverse is unique.The inverse of A is denoted by .

1A IAAAA 11


Proof of Theorem 2.7Proof of Theorem 2.7 Because A is invertible, it has at least one inverse B such

that AB = BA = I. Suppose that A has another inverse C such that

AC = CA = I. Then you can show that B and C are equal as follows.

AB = I C(AB) = C(I) (CA)B = C (I)B = C B = C

Consequently B = C, and it follows that the inverse of a matrix is unique.

Section 2-3


Example 2Example 2Find the inverse of the matrix Sol: To find the inverse of A,

try to solve the matrix equation AX = I for X.

31

41A

1,413

04

1,303

14

10

01

33

44

10

01

31

41

22122212

2212

21112111

2111

22122111

22122111

2221

1211

xxxx

xx

xxxx

xx

xxxx

xxxx

xx

xx

11

431 XA

Using matrix multiplication to check the result.

10

01

11

43

31

41

Section 2-3


Gauss-Jordan EliminationGauss-Jordan Elimination

13

04

03

14

2212

2212

2111

2111

xx

xx

xx

xx

031

141

131

041 The same coefficient matrix

Double augment matrix

1031

0141

1110

0141

(4)

1110

4301

[ A ┇ I ] … [ I ┇ A1 ]

Section 2-3


ProcedureProcedureLet A be a square matrix of order n. Write the n2n matrix [ A┇I ] (adjoining the matrices A & I) If possible, row reduce A to I using elementary row

operations on the entire matrix [ A┇I ].The result will be the matrix [I ┇A1 ].If this in not possible, then A is not invertible.

Check your work by multiplying to see thatAAIAA 11

Section 2-3


Example 4Example 4 Show that the matrix has no inverse.

pf:

232

213

021

A

100232

010213

001021

IA

102270

013270

001021

111000

013270

001021

(3)

2

It is not possible to rewrite [A┇I ] in the form [I ┇A1 ].

Hence A has no inverse.

Section 2-3


The Inverse of a 2The Inverse of a 22 Matrix2 Matrix The matrix A is a 22 matrix given by The matrix A is invertible if and only if

= ad bc 0. If 0, then

pf:

dc

baA

ac

bdA

11

10

01

0

01

11

bcad

bcad

bcad

ac

bd

dc

ba

bcadAA

Interchanging the entires on the main diagonal and changing the signs of the other two entires.

Section 2-3

+


Example 5Example 5 If possible, find the inverse of each matrix.

22

13)( Aa

04 bcad

43

21

41

21

1

32

12

4

1A

26

13)( Bb

0)6)(1()2)(3( bcad

The matrix B is not invertible.

Section 2-3


Theorem 2.8Theorem 2.8 Properties of Inverse Matrix

If A is an invertible matrix, k is a positive integer, and c is a scalar, then A1, Ak, cA, and AT are invertible and the flowing are true.1. 2.3. 4.

Hint: if BC = CB = I, then C is the inverse of B.

pf: 1. Observe that , which means that A is the inverse of A1. Thus, .

3.

Section 2-3

AA 11)( kk AA )()( 11 0,)( 111 cAcA c

TT AA )()( 11

IAAAA 11

AA 11)(

IIAAccAA

IIAAcAcA

cc

cc

)1())(())((

)1())(())((1111

1111 Hence is the inverse of (cA), which implies that

11 AC

0,)( 111 cAcA c


Example 6Example 6Compute A2 in two different ways and show that the results

are equal.1. (A2)1:

2. (A1)2:

42

11A

1810

53

42

11

42

112A 4)10)(5()18)(3(

43

25

45

29

12

310

518

4

1)(A

2)1)(2()4)(1(42

11

A

21

21

1

1

2

12

14

2

1A

43

25

45

29

21

21

21

21

21

1

2

1

2)(A

the same result

Section 2-3


Theorem 2.9Theorem 2.9 The Inverse of a Product

If A and B are invertible matrices of size n,then AB is invertible and (AB)1 = B1A1.

pf: 1. (AB)(B1A1) = A(BB1)A1 = A(I)A1 = AA1 = I. 2. (B1A1)(AB) = B1(A1A)B = B1(I)B = B1B = I. Hence AB is invertible.

: in reverse order

Section 2-3

Recall: (AB)T = BTAT

11

12

1121 )( AAAAAA nn


Example 7Example 7Find (AB)1 for the matrices and

using the fact that A1 and B1 are given by

Sol:

431

341

331

A

342

331

321

B

101

011

3371A

31

32

1

0

011

121

B

37

31

32

111

25

348

258

101

011

337

0

011

121

)( ABAB

242712

242611

212310

AB

3725100

348010

258001

100242712

010242611

001212310

Section 2-3


Theorem 2.10Theorem 2.10 Cancellation PropertyIf 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: Use that fact that C is invertible and writeAC = BC (AC)C1 = (BC)C1

A(CC1) = B(CC1)

AI = BI

A = B

Section 2-3


Theorem 2.11Theorem 2.11 Systems of Equations with Unique Solutions

If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given byx = A1b.

pf: Ax = b A1(Ax) = A1(b) A1Ax = A1b x = A1b Example 8: Use an inverse matrix to solve each system

142133132)(

zyxzyxzyxa

042033032)(

zyxzyxzyxc

326101011

142133132

1AA

212

211

326101011

)( 1bx Aa

000

)( 1bx Ac

Section 2-3


2.4 Elementary Matrices2.4 Elementary Matrices Definition: An nn matrix is called an elementary

matrix if it can be obtained from the identity matrix In by a single elementary row operation.

Elementary row operations:1. Interchange two rows.2. Multiply a row by a nonzero constant.3. Add a multiple of a row to another row.

The identity matrix In is elementary. it can be obtained from itself by multiplying any one of its row by 1.


Example 1Example 1 Which matrices are elementary?

000010001

)(010001

)(100030001

)( cba

100020001

)(1201

)(010100001

)( fed

: two elementary row operations are required.

: (3)R2 R2: it is not a square matrix

: (0)R3 R3

must be by a nonzero const.

: R2 R3 : R2+R1 R2

Section 2-4


Example 2Example 2Elementary Matrices & Elementary Row Operations Interchange two rows: R1 R2

Multiply a row by a nonzero constant: (0.5)R2 R2

Add a multiple of a row to another row: R2+(2)R1 R2

123120631

123631120

100001010

131023101401

131046201401

10000001

21

540120101

540322101

100012001

Section 2-4


Theorem 2.12Theorem 2.12Representing Elementary Row Operations

Let E be the elementary matrix obtained by performing an elementary row operation on Im. If the same elementary row operation is performed on an mn matrix A, then the resulting matrix is given by the product EA.

Most applications of elementary row operations require a sequence of operations. Gaussian elimination

Section 2-4


Example 3Example 3Find a sequence of elementary matrices that can be used to

write the matrix A in row-echelon form.

Sol:

026220315310

A

026253102031

100001010

1E

The elementary matrix E is a 33 matrix.

(2)

420053102031

102010001

2E

(1/2)

210053102031

21

3

00010001

E

AE1

)( 12 AEE

)( 123 AEEE

Section 2-4


Row Equivalence & Thms. 2.13~14Row Equivalence & Thms. 2.13~14 Definition: Let A and B be mn matrices. Matrix B is

row-equivalent to A if there exists a finite number of elementary matrices E1, E2, …, Ek such thatB = EkEk1E2E1A.

Theorem 2.13: Elementary Matrices Are InvertibleIf E is an elementary matrix, then E1 exists and is an elementary matrix.

Theorem 2.14: A Property of Elementary MatricesA square matrix A is invertible if and only if it can be written as the product of elementary matrices.

Section 2-4


Elementary Matrices Are InvertibleElementary Matrices Are Invertible

Elementary Matrix Inverse Matrix

21

3

2

1

00

010

001

102

010

001

100

001

010

E

E

E

200

010

001

102

010

001

100

001

010

13

12

11

E

E

ER1 R2

R3+(2)R1 R3

(½)R3 R3

R1 R2

R3+(2)R1 R3

(2)R3 R3

Section 2-4


Proof of Theorem 2.14Proof of Theorem 2.14() If A can be written as the product of elementary matrices, then

A is invertible.pf: Assume that A is the product of elementary matrices. Then,

because every elementary matrix is invertible and the product of invertible matrices is invertible, it follows that A is invertible.

() If A is invertible, then it can be written as the product of elementary matrices.

pf: Assume that A is invertible. The system of linear equationsAX = O has only the trivial solution. This implies that [A┇O] can be rewritten in the form [I┇O] using the elementary operations as EkEk1E2E1A = I. Then it follows that . Thus A can be written as the product of elementary matrices.

112

11

kEEEA

Section 2-4


Example 4Example 4Find a sequence of elementary matrices whose product is

Sol:

83

21A

(1)

10

01

83

211E(3)

13

01

20

212E

213 0

01

10

21E

(½)

(2)

10

21

10

014E

IAEEEE 12341

41

31

21

1 EEEEA

10

0111E

13

0112E

20

0113E

10

2114E

Section 2-4


Theorem 2.15Theorem 2.15 Equivalent Conditions

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 vector b.

3. Ax = O has only the trivial solution.

4. A is row-equivalent to In.

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

Section 2-4


LU-FactorizationLU-Factorization A square matrix A is expressed as a product A = LU, where

the square matrix L is lower triangular and the square matrix U is upper triangular.

Step 1: EkE2E1A = U : row reduction Step 2: Step 3: A = LU

33

2322

131211

333231

2221

11

00

0,0

00

a

aa

aaa

U

aaa

aa

a

LBy row reducing A

Section 2-4

UEEEA k11

21

1

112

11

kEEEL


Example 6Example 6Find the LU-factorization of the matrix

Sol:

2102

310

031

A

Section 2-4

(2)

102

010

001

240

310

031

1E4

140

010

001

1400

310

031

2E

102

010

0011

1E

104

010

0011

2E

UAEE 12 UEEA 12

11

= U

142

010

0011

21

1 EE = L LU = A


Solving a Linear SystemSolving a Linear System Using LU-factorization to solve the linear system Ax = b:

Ax = b and A = LU LUx = bLet Ux = y. Ly = b1. Solve Ly = b for y. (forward substitution)2. Solve Ux = y for x. (back-substitution)

Section 2-4


Example 7Example 7Solve the linear system.Sol:

1. Let y = Ux and solve Ly = b for y

2. Solve Ux = y for x

202102

13

53

321

32

21

xxx

xx

xx

1400

310

031

142

010

001

2102

310

031

A

20

1

5

142

010

001

3

2

1

y

y

y

14

1

5

3

2

1

y

y

y

y

14

1

5

1400

310

031

3

2

1

x

x

x

1

2

1

3

2

1

x

x

x

x

Section 2-4

chap. 2 matrices 2.1 operations with matrices 2.2 properties of matrix operations 2.3 the inverse of...

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