chapter 2 matrices

CHAPTER 2

MATRICES

2.1 Operations with Matrices2.2 Properties of Matrix Operations2.3 The Inverse of a Matrix2.4 Elementary Matrices2.5 Applications of Matrix Operations

Elementary Linear Algebra 投影片設計編製者R. Larson (7 Edition) 淡江大學電機系翁慶昌教授

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CH 2 Linear Algebra Applied

Flight Crew Scheduling (p.47) Beam Deflection (p.64)

Information Retrieval (p.58)

Computational Fluid Dynamics (p.79) Data Encryption (p.87)

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2.1 Operations with Matrices

Matrix:

nm

nmmnmmm

n

n

n

nmij M

aaaa

aaaa

aaaa

aaaa

aA

][

321

3333231

2232221

1131211

(i, j)-th entry: ija

row: m

column: n

size: m×n

Elementary Linear Algebra: Section 2.1, p.40

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i-th row vector

iniii aaar 21

j-th column vector

mj

j

j

j

c

c

c

c2

1

row matrix

column matrix

Square matrix: m = n


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Diagonal matrix:

),,,( 21 nddddiagA nn

n

M

d

d

d

00

00

00

2

1

Trace:

nnijaA ][ If

nnaaaATr 2211)(Then

Elementary Linear Algebra: Section 2.1, Addition

6/75

Ex:

654

321A

2

1

r

r

321 ccc

,3211 r 6542 r

,4

11

c ,

5

22

c

6

33c

654

321A


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nmijnmij bBaA ][ ,][ If

Equal matrix:

njmibaBA ijij 1 ,1 ifonly and if Then

Ex 1: (Equal matrix)

dc

baBA

43

21

BA If

4 ,3 ,2 ,1 Then dcba


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Matrix addition:

nmijnmij bBaA ][ ,][ If

nmijijnmijnmij babaBA ][][][Then

Ex 2: (Matrix addition)

31

50

2110

3211

21

31

10

21

2

3

1

2

3

1

22

33

11

0

0

0


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Matrix subtraction:

BABA )1(

Scalar multiplication:

scalar : ,][ If caA nmij

Ex 3: (Scalar multiplication and matrix subtraction)

212

103

421

A

231

341

002

B

Find (a) 3A, (b) –B, (c) 3A – B

nmijcacA ][Then


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(a)

212

103

421

33A

(b)

231

341

002

1B

(c)

231

341

002

636

309

1263

3 BA

Sol:

636

309

1263

231323

130333

432313

231

341

002

407

6410

1261


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Matrix multiplication:

pnijnmij bBaA ][ ,][ If

pmijpnijnmij cbaAB ][][][Then

njin

n

kjijikjikij babababac

1

2211where

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

Size of AB

inijii

nnnjn

nj

nj

nnnn

inii

n

cccc

bbb

bbb

bbb

aaa

aaa

aaa

21

1

2221

1111

21

21

11211

Elementary Linear Algebra: Section 2.1, p.42 & p.44

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05

24

31

A

14

23B

Ex 4: (Find AB)

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


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Matrix form of a system of linear equations:

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

2211

22222121

11212111

= = =

A x b

equationslinear m

equationmatrix Single

bx A 1 nnm 1m

mnmnmm

n

n

b

b

b

x

x

x

aaa

aaa

aaa

2

1

2

1

21

22221

11211


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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


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n

mnmm

n

n

ccc

aaa

aaa

aaa

A

21

21

22221

11211

nx

x

x

x2

1

12211

2222121

1212111

mnmnmm

nn

nn

xaxaxa

xaxaxa

xaxaxa

Ax

mn

n

n

n

a

a

a

x

2

1

1

21

11

1

ma

a

a

x

2

22

12

2

ma

a

a

x

1c 2c nc


Linear combination of column vectors:

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9

6

3

,

8

5

2

,

7

4

1

,

6

3

0

, ,

987

654

321

321

3

2

1

cccb

x

x

x

xA

Ex 7 ： (Solve a system of linear equations)

bxxx

xxx

xxx

xxx

Ax

6

3

0

9

6

3

8

5

2

7

4

1

987

654

32

321

321

321

321

(infinitely many solutions)6877

3654

032

321

321

321

xxx

xxx

xxx

)1 ,1 ,1 i.e. ,

1

1

1

:solution (one

6

3

0

9

6

3

)1(

8

5

2

1

7

4

1

1 321

xxxx


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Key Learning in Section 2.1

Determine whether two matrices are equal.

Add and subtract matrices and multiply a matrix by a scalar.

Multiply two matrices.

Use matrices to solve a system of linear equations.

Partition a matrix and write a linear combination of column vectors..

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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: 純量乘法 ( 純量積 )

matrix subtraction: 矩陣相減 matrix multiplication: 矩陣相乘 partitioned matrix: 分割矩陣 linear combination: 線性組合

Keywords in Section 2.1

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2.2 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

Elementary Linear Algebra: Section 2.2, 52-55

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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:


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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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(1) A(BC) = (AB)C

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

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

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

Properties of identity matrix:

AAI nmMA

n

)1(Then

If

AAI m )2(

Properties of matrix multiplication:

Elementary Linear Algebra: Section 2.2, p.54 & p.56

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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

Transpose of a matrix:


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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)

Ex 8: (Find the transpose of the following matrix)


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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:


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0

30

210

If

cb

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

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:


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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:

BAAB pnnm

Three situations:


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12

31A

20

12B

Sol:

44

52

20

12

12

31AB

Note: BAAB

24

70

12

31

20

12BA

Ex 4 :

Sow 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, 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

Ex 5: (An example in which cancellation is not valid)

Show that AC=BC


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Use the properties of matrix addition, scalar multiplication, and zero matrices.

Use the properties of matrix multiplication and the identity matrix.

Find the transpose of a matrix.

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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


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2.3 The Inverse of a Matrix

nnMA

,such that matrix a exists thereIf nnn IBAABMB

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. Notes:(1) The inverse of A is denoted by 1A

IAAAA 11 )2(

Thm 2.7: (The inverse of a matrix is unique)

Elementary Linear Algebra: Section 2.3, pp.62-63

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1nEliminatioJordan -Gauss || AIIA

Ex 2: (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 by Gauss-Jordan Elimination:


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110301

031141)1(

)4(21

)1(12 ,

rr

110401

131041)2(

)4(21

)1(12 ,

rr

1 ,3 2111 xx

1 ,4 2212 xx

)( 11

43 1-

2221

12111 AAIAXxx

xxAX

Thus

(2) 1304

(1) 0314

2212

2212

2111

2111

xxxx

xxxx


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1110

4301

1031

0141

1

inationJordanElimGauss

, )4(21

)1(12

AIIA

rr

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

Note:


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326101011

A

100326011110001011

)1(

12

r

Sol:

100010001

326101011

IA

106011001

340110011

)6(

13

r

142100011110001011

)1(

3

r

142100011110001011

)4(

23

r

Ex 3: (Find the inverse of the following matrix)


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142100133010001011

)1(

32

r

141100133010132001

)1(

21

r

So the matrix A is invertible, and its inverse is

142

133

1321A

] [ 1 AI

Check:

IAAAA 11


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IA 0(1)

0)( )2(factors

kAAAAk

k

integers:, )3( srAAA srsr rssr AA )(

kn

k

k

k

n d

d

d

D

d

d

d

D

00

00

00

00

00

00

)4( 2

1

2

1

Power of a square matrix:


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

not equal to zero, then

AAA 111 )( and invertible is (1)

kk

k

kk AAAAAAA )()( and invertible is (2) 1

factors

1111

0 ,1

)( and invertible is c (3) 11 cAc

cAA

TTT AAA )()( and invertible is (4) 11

Thm 2.8 ： (Properties of inverse matrices)


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Thm 2.9: (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:

11

12

13

11321

AAAAAAAA nn


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Thm 2.10 (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.


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Thm 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 by

bAx 1Pf:

( 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)


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bAx

bAIbAAAbA A 111 |||1

Note:


(A is an invertible matrix)

Note:

For square systems (those having the same number of equations

as variables), Theorem 2.11 can be used to determine whether the

system has a unique solution.

nA

n bAbAIbbbA 11

121 |||||||

1

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▪ Find the inverse of a matrix (if it exists).▪ Use properties of inverse matrices.▪ Use an inverse matrix to solve a system of linear equations.

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


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2.4 Elementary Matrices

Row elementary matrix:

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

obtained

from the identity matrix In by a single elementary operation. Three row elementary matrices:

)( )1( IrR ijij

)0( )( )2( )()( kIrR ki

ki

)( )3( )()( IrR kij

kij

Interchange two rows.

Multiply a row by a nonzero constant.

Add a multiple of a row to another row.

Note:

Only do a single elementary row operation.


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100030001

)(a

010001)(b

000010001

)(c

010100001

)(d

1201)(e

100020001

)( f

))(( Yes 3(3)

2 Ir square)(not Not)constan nonzero aby bemust

tionmultiplica (Row No

))(( esY 323 Ir ))(( esY 2(2)

12 Ir)operations row

elementary two(Use No

Ex 1: (Elementary matrices and nonelementary matrices)


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Notes:

ARAr ijij )( )1(

ARAr ki

ki

)()( )( )2(

ARAr kij

kij

)()( )( )3(

EAAr

EIr

)(

)(

Thm 2.12: (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.


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))((

123

120

631

123

631

120

100

001

010

)( 1212 ARAra

Ex 2: (Elementary matrices and elementary row operation)


))((

540

120

101

540

322

101

100

012

001

)( )5(12

)2(12 ARArc

))((

1310

2310

1401

1310

4620

1401

100

02

10

001

)()

2

1(

2

)2

1(

2 ARArb

53/75

0262

2031

5310

A

Sol:

100

001

010

)( 3121 IrE

102

010

001

)( 3)2(

132 IrE

21

3

)2

1(

33

00

010

001

)(IrE

Ex 3: (Using elementary matrices)

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

the matrix A in row-echelon form.


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026253102031

026220315310

100001010

)( 1121 AEArA

420053102031

026253102031

102010001

)( 121)2(

132 AEArA

210053102031

420053102031

2

100

010001

)( 232

)2

1(

33 AEArA

AEEEB 123 )))(((or 12)2(

13

)2

1(

3 ArrrB

B

row-echelon form


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

of elementary matrices such that

AEEEEB kk 121

Row-equivalent:


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Thm 2.13: (Elementary matrices are invertible)

If E is an elementary matrix, then exists and

is an elementary matrix.

Notes:

ijij RR 1)( )1(

)1

(1)( )( )2( ki

ki RR

)(1)( )( )3( kij

kij RR

1E


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121100001010

RE

100001010

)( 11

112 ER

)2(132

102010001

RE

102010001

)( 12

1)2(13 ER

)2

1(

3

21

300

010001

RE

200010001

)( 13

1)2

1(

3 ER

Ex:

Elementary Matrix Inverse Matrix


12R (Elementary Matrix)

)2(13R (Elementary Matrix)

)2(3R (Elementary Matrix)

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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. (Thm. 2.11)

0xA 00 IA

IAEEEEk 123 11

31

21

1 kEEEEA

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

Thm 2.14: (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.


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83

21A

Sol:

I

A

rr

rr

10

01

10

21

20

21

83

21

83

21

)2(21

)2

1(

2

)3(12

)1(1

IARRRR )1(1

)3(12

)2

1(

2)2(

21 Therefore

Ex 4:

Find a sequence of elementary matrices whose product is


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1)2(21

1)2

1(

21)3(

121)1(

1 )()()()( Thus RRRRA)2(

21)2(

2)3(

12)1(

1 RRRR

10

21

20

01

13

01

10

01

Note:

If A is invertible

][][ 1123

AIIAEEEEk

IAEEEEk 123 Then

1231 EEEEA k


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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 2.15: (Equivalent conditions)


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LUA L is a lower triangular matrix

U is an upper triangular matrix

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

lower

triangular matrix L and an upper triangular matrix U, then

A=LU is an LU-factorization of A

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 row to another row below it, then it is easy to find an LU-

factorization of A.

LUA

UEEEA

UAEEE

k

k

11

21

1

12

LU-factorization:


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01

21 )( Aa

2102310031

)( Ab

UA r

20

21 0121 (-1)

12

Sol: (a)

UAR )1(12

LUURA 1)1(12 )(

11

01)( )1(

121)1(

12 RRL

Ex 5: (LU-factorization)


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(b)

UA rr

1400

310

031

240

310

031

2102

310

031)4(

23)2(

13

UARR )2(13

)4(23

LUURRA 1)4(23

1)2(13 )()(

142010001

140010001

102010001

)()( )4(23

)2(13

1)4(23

1)2(13 RRRRL


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bLUxLUA then ,If

bLyUxy then , Let

Two steps:

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

(2) Solve Ux = y for x

bAx

Solving Ax=b with an LU-factorization of A


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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 7: (Solving a linear system using LU-factorization)


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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


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Factor a matrix into a product of elementary matrices.

Find and use an LU-factorization of a matrix to solve a system of linear equations.

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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


70/75


Write and use a stochastic matrix.

Use matrix multiplication to encode and decode messages.

Use matrix algebra to analyze an economic system (Leontief input-output model).

Find the least squares regression line for a set of data.

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Fight Crew Scheduling

Many real-life applications of linear systems involve enormous numbers of equations and variables. For example, a flight crew scheduling problem for American Airlines required the manipulation of matrices with 837 rows and more than 12,750,000 columns. This application of linear programming required that the problem be partitioned into smaller pieces and then solved on a Cray supercomputer.

2.1 Linear Algebra Applied


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Information Retrieval

Information retrieval systems such as Internet search engines make use of matrix theory and linear algebra to keep track of, for instance, keywords that occur in a database. To illustrate with a simplified example, suppose you wanted to perform a search on some of the m available keywords in a database of n documents. You could represent the search with the m1 column matrix x in which a 1 entry represents a keyword you are searching and 0 represents a keyword you are not searching. You could represent the occurrences of the m keywords in the n documents with A, an m n matrix in which an entry is 1 if the keyword occurs in the document and 0 if it does not occur in the document. Then, the n1 matrix product ATx would represent the number of keywords in your search that occur in each of the n documents.



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Beam Deflection

Recall Hooke’s law, which states that for relatively small deformations of an elastic object, the amount of deflection is directly proportional to the force causing the deformation. In a simply supported elastic beam subjected to multiple forces, deflection d is related to force w by the matrix equation

d = Fw

where is a flexibility matrix whose entries depend on the material of the beam. The inverse of the flexibility matrix, F‒1 is called the stiffness matrix. In Exercises 61 and 62, you are asked to find the stiffness matrix F‒1 and the force matrix w for a given set of flexibility and deflection matrices.



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Computational Fluid Dynamics

Computational fluid dynamics (CFD) is the computer-based analysis of such real-life phenomena as fluid flow, heat transfer, and chemical reactions. Solving the conservation of energy, mass, and momentum equations involved in a CFD analysis can involve large systems of linear equations. So, for efficiency in computing, CFD analyses often use matrix partitioning and -factorization in their algorithms. Aerospace companies such as Boeing and Airbus have used CFD analysis in aircraft design. For instance, engineers at Boeing used CFD analysis to simulate airflow around a virtual model of their 787 aircraft to help produce a faster and more efficient design than those of earlier Boeing aircraft.



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Data Encryption

Because of the heavy use of the Internet to conduct business, Internet security is of the utmost importance. If a malicious party should receive confidential information such as passwords, personal identification numbers, credit card numbers, social security numbers, bank account details, or corporate secrets, the effects can be damaging. To protect the confidentiality and integrity of such information, the most popular forms of Internet security use data encryption, the process of encoding information so that the only way to decode it, apart from a brute force “exhaustion attack,” is to use a key. Data encryption technology uses algorithms based on the material presented here, but on a much more sophisticated level, to prevent malicious parties from discovering the key.



chapter 2 matrices

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