8.1 vector spaces a set of vector is said to form a linear vector space v chapter 8 matrices and...

8.1 Vector spaces

A set of vector is said to form a linear vector space V

Chapter 8 Matrices and vector spaces

..... , , cba

tindependen linearly are ,......,,0.... If (II)

dependent linearly are ,....,,zero allnot ,...,,, If (I)

0.......

0)(vector negative ingcorrespond a have vectors All(5)

unity is 11 (4)

vector anyfor 00vector null a exists There (3)

scalars arbitrary are , )()(

)(

)( (2)

)()( , (1)

scba

scba

sbaαX

aaa

aa

aaa

aa

aaa

baba

cbacbaabba


Basis vector

i

N

iiN

N

N

NN

exXVeeee

xxxx

eeee

exexexexX

VX

V

1321

321

321

332211

for basis a form ,...,, then

zero to equal allnot are ,......,,

and t,independen linearly are,........,,

......... if

invector arbitrary an is

spacevector arbitrary an is

The inner product is defined by ba

baba

cbcabcaccbcacba

cabacba

abba

bababa

*

****

*

(5)

(4)

(3)

(2)

cos||||space ldimensiona-three real In (1)


some properties of inner product

baeebaeeba

ebebebeaeaeaba

aaeeaeaeae

ebbeaa

ji

jiδ

eeeee

aaaaa

aaaa

baba

i

N

iiji

N

i

N

ijjiji

N

iii

NNNN

j

N

iijiij

N

ii

N

iiijj

N

iii

N

iii

ij

ijji

1

*

1

*

1

*

22112211

111

11

321

2/1

ˆˆˆˆ

ˆ.....ˆˆˆ.....ˆˆ

ˆˆˆˆˆ

ˆ and ˆ (4)

for 0

for 1 deltaKronecker

lorthonorma is ,....ˆ,ˆ,ˆ basis aˆˆ if (3)

00 if0

is of norm (2)

orthogonal are and 0 (1)


real is

(6)

ˆˆˆˆ

ˆˆˆˆ

ˆˆ define

lorthonormanot is ˆ,.....ˆ,ˆ,ˆ basis a if (5)

1/2

11 1

**

1 1

*1

*

**

1 1

*

1 1

*

11

321

aaa

aaaGaaGaaa

Geeee G

bGaeebaebeaba

eeG

eeee

N

jji

N

ijj

N

i

N

jij

jiijjiij

N

ijij

N

jiji

N

i

N

jji

N

jjj

N

iii

jiij

N

222

2

222

22

2222

2

*2222

**2

||

04

)4||4()

||(

||20

choose , allfor 0for

|||||||| if

:Proof

scalar a is , when hold, equality the

inequality sSchwarz' (1)

baba

b

baba

b

bar

barbraba

reba

ebaebababaebaba

bbabbaaabababa

bab a|ba |

i

iii

Some useful inequalities:


baba

bababababa

bababa

baba

)(2||2

Re2

:Proof

inequality triangle the (2)

22222

222


|ˆ|ˆˆˆˆˆˆ2

ˆˆˆˆˆˆ 0

:Proof

|||ˆ|,....2,1 ,ˆ basis lorthonorma a

inequality sBessel' (3)

22

2

1

22

1

2

iijij

i jii

ii

jjj

iii

iii

N

ii

N

iii

aeaeeaeeaaeeaaa

eaeaeaeaeaea

aaeaNie

)(2

22

:Proof

)(2

equality ramparallelog the (4)

22

22

2222

ba

abbabbabbaaa

babababababa

bababa


'

1

''

'

1

'

1

''

1

'

'

1

1111

1

component

)(

,...3,2,1 , basis new afor

component

)(

,....3,2,1 , basis afor (3)

scalars are , )( (2)

to transfers (1)

operatorlinear a is

j

N

jiji

i

N

iij

N

jji

N

ii

i

j

N

jiji

N

iiij

N

jj

N

jjji

N

ii

i

N

iijji

xAy

eAxAxAeyy

Nie

xAy

eAxexAxAeyy

eAeANie

bAaAbaA

yxAxAy

A


8.2 Linear operators

The action of operator A is independent of any basis or coordinate system.

i

M

iijj

i

i

fAeA

MifNMy

Niex

1

,....2,1 , basis , spacevector ldimensiona-M in

,....2,1 , basis spacevector ldimensiona-N in


Properties of linear operators:

1operator inverse *

equal are and vector allfor *

1operator identity *

0operator null *

)()(

)()(

)(

11

AAAA

BAxxBxA

xx

xO

xBAxAB

xBAxAB

xAxA

xBxAxBA

A


8.3 Matrices

x

y

spacevector

ensionalN

dimNj

e j

,..2,1

basis

Mi

f i

,...2,1

basis

space

dim

vector

ensionalM

,......N, ieNie

xxx

x

x

x

xxxx

x

x

x

x

AAAA

AAA

AAA

A

ii

T

NN

MMMM

N

N

21, basis in ,...2,1 , basis in

Vector Vector

matrix NM ofMatrix

'

'3

'2

'1

'

'2

'1

'T421

2

1

4321

22221

11211

kjk

ikij

ijij

ijijij

jjjk

kikk

kikj

jij

jj

ijjj

ij

ijj

ijjj

ijjj

ij

BAAB

AA

BABA

xBABxAxAB

xAxA

xxBxAxBAxxBA

)( (6)

)( (5)

)((4)

)()( (3)

)( (2)

)()( (1) ''


8.4 Basic matrix algebra

Matrix addition ABBASBAS ijijij

Multiplication by a scalar

2221

1211

2221

1211

AA

AA

AA

AA


Multiplication of matrices

Mj

j

j

MNMM

N

N

j

j

NN

NMNMM

N

N

M

j

N

jiji

A

A

A

AAA

AAA

AAA

Ae

eA

xAxAxAy

x

x

x

AAA

AAA

AAA

y

y

y

Axy

NixAyxAy

2

1

21

22221

11211

22221212

2

1

21

22221

11211

2

1

1

0

1

0

vector basis a on operates Operator (2)

.........

formmatrix

,.....2,1for (1)

100

010

001

1

111matrix Identity (8)

and 0matrix zeroor Null (7)

)(

)(

additionover veDistributi (6)

ecommutativ-Non (5)

)()( ve Associati(4)

RM : R N : NM :

Matrix (3)

3

3

132

22

12

31

21

11

23

13

22

12

21

11

2221

1211

AAA

AAOOAOAAOO

CBCABAC

BCACCBA

BAAB

CABBCA

BAP

B

B

B

B

B

B

A

A

A

A

A

A

PP

PP

PBA

ABP

kkjikij


0 !)exp( function A

...... definematrix square aFor

n

nn

nn

n

n

AAAaS

AAAAAA


Functions of matrices

The transpose of a matrix

TTTTT

ijTT

kjT

ikk

T

ikT

kjk

Tkj

kjkji

Tij

TTT

Tjiij

T

T

ABCGGABC

ABAB

BABAABAB

ABAB

AAAA

AA

.........).....(

)()()(

)()()()( :Proof

)(

1

4

0

2

1

3

140

213 Ex. )(

matrix a of transpose the is

A


For a complex matrix NM

ABGGAB

i

i

Ai

iA

A AA

AAA

AAA

AA

TT

ijij

.........).........(

0

1

1

3

2

1

011

321 Ex.

real is if

)()( (adjoint) conjugate Hermitian

real is if

)()( conjugateComplex

**

*

**

productinner 1

*2

1

**2

*1

2

1

2

1

baba

b

b

b

aaaba

b

b

b

b

a

a

a

aN

iii

N

N

NN


If the basis is not orthonormal ie

GbabGabeeaebeaba

ebbeaaGee

jij

N

i

N

jijji

N

i

N

jij

N

i

N

jjii

i

N

iii

N

iiijijji

1 1

*

1 1

*

1 1

*

11

, and

The trace of a matrix

*

11 11 11

12211

)( and (4)

)()()( (3)

)()()()( (2)

)()()( (1)

.....

TrATrATrATrA

CABTrBCATrABCTr

BATrBAABBAABABTr

BTrATrBATr

AAAATrA

T

N

jjjij

N

i

N

jji

N

i

N

jjiij

N

iii

N

iiiNN

row 1th the by expanded

)1()1()1(

)1()1()1(

||det

1

3

11131312121111

1331

131221

121111

11

3231

22213113

3331

23212112

3332

23221111

333231

232221

131211

jj

jCACACACA

MAMAMA

AA

AAA

AA

AAA

AA

AAA

AAA

AAA

AAA

AA


8.9 The determinant of a matrix

column ith the by expanded ...,3,2,1

row ith the by expanded ...,3,2,1 ||

)1( :Cofactor

of column jth and row ith the of elements the all removing bymatrix

)1()1( the oft determinan the is )(element the of :Minor

1

1

NiCA

NiCAA

MC

A

NNNNAM

ji

N

jji

ij

N

jij

ijji

ij

ijij


Ex: Matrix A is 3×3, for three 3-D vectors

),,( ),,,( ),,,( 321321321 ccccbbbbaaaa cba

, ,

cba

A

cbacbcbacbcbacbcba

ccc

bbb

aaa

A

and ,

vectors the by define ipedparallelep the of volume the is ||

)()(()(

||

122133113223321

321

321

321

Properties of determinants

magnitude. in unaltered isbut sign changest determinan its

:columns twoor rows two ingInterchang (3)

|||||)(|||||| |

(adjoint) conjugate Hermitian andcomplex (2)

transpose (1)

***** AAAAAA

|A| | |A

T

T


npermutatio cyclicfor invariant is

|.....||.....||||||.....|

|.....||.....||||||......||...........|

||||||||

Product (7)

unchanged is ||another to (column) row one of multipleconstant a Adding(6)

0|| columnsor rows Identical (5)

||||

of (column) row single a infactor common a is matrix, NN A

factors Removing (4)

FGABGABFFGAB

FGABFABGGABFGAB

BABAAB

A

A

AA

AN

row 3th the to lpropotiona is row 1th the 0

202

113

404

2

202

113

311

)1(12

2012

1133

0010

3101

2

1012

2133

1010

3101

2

1112

2233

1110

3101

2

1212

2433

1210

3201

||

22

A


Ex: Evaluate the determinant

(2)+(3) put (3)

(4)-(2) put (4)

BABA

IAAAAIAAI

PBAPABPAB

1

11

11

matrixunit the is

and

The elements of the inverse matrix are

1A


8.10 The inverse of a matrix

kik

kiijkjk

ki

ijkjk

kikjik

kij

kikikiik

T

ik

ACAAAC

A

AA

A

CAAAA

ACA

C

A

CA

||||for

||

||

||)()( :Proof

ofcofactor the is ||||

)()(

11

1

jiCACAA

aaa

aaa

Aaa

aaa

A

kik

kjkik

ki

jj

jj

i

ji

for 0||

..................

......

......

..................

...........

......

'''

2221

1111'

221

1111

1111111

1111111111

111111

111

1111**1*1

11

11111

11

11111111

11

...........)(.........)( (5)

)()(

)())(())((

)( (4)

)()()())(())(())((

)()( (3)

)()()()()(

)()( (2)

)()(1)(

)( (1)

ABCGABCGABC

ABABABBBABABBB

AABBAABABAIABAB

ABAB

AAAAAA

AA

AAAAAAIIAA

AA

AAAAAAAA

AA

TTT

TTTTTTTT

TT


Useful properties:

1**1*1***1

**11

111

)()()()(

)()(

:Proof

||/1||1||||||||

AAAAIAA

IAAIAA

AAIAAAA


8.12 Special types of square matrix

Diagonal matrices

matrice diagonal are both and If

/1....0

.........

0.../10

0...0/1

........||

22

11

1

1332211

BAABBA

A

A

A

A

AAAAAA

NN

N

iiiNN


Lower triangular matrix

NNAAAA

AAA

AA

A

A ......||0

00

1211

333231

2221

11

Upper triangular matrix

NNAAAA

A

AA

AAA

A .....||

00

0 2211

33

2322

131211

Symmetric and antisymmetric matrices

ACCAAC

ABBAAB

CBAAAAA

AAAAAAA

AAAA

AAAA

TT

TT

TT

TTT

jiijT

jiijT

ofpart ricantisymmet 2/)(

ofpart symmetric 2/)(

2/)(2/)( *

symmetry. same the have and )()( If

:ricAntisymmet

:Symmetric

1111


Ex: If A is N×N antisymmetric matrix, show that |A|=0 if N is odd.

0||||||odd is if

||)1(||||||

AAAN

AAAAAA NTT

Orthogonal matrices

xxxxAxAxyyyy

Axy

AIAAAAA

AAAA

AAA

TTTT

TT

TT

T

unchanged. isvector real a of norm The (3)

1||1|||||||||| (2)

orthogonal also is )()()( (1)

matrix orthogonal an is If

2

11111

1

Hermitian and anti-Hermitian matrices

ACCAAC

ABBAAB

CBAAAAA

AAA

AAAA

ofpart Hermitian-anti 2/)(

ofpart Hermitian 2/)(

2/)(2/)( )2(

)()( (1)

Hermitian-anti Hermitian 111


Unitary matrices

unchanged. isvector complex a of norm The

)()(

and if (3)

1|||||||||||| (2)

unitary also is )()()( (1)

then real is if

1

*

11111

1

xxxxAxAxAxAxyyyy

AAAxy

IAAAAAA

AAAA

AAAAA T

Normal matrices

AAAA

normal is inverse itsnormal is if

)()()()())(( (2)

:Unitary

:Hermitian

matrices normal are matrices unitary and matrices Hermitian (1)

1

111111111

111

AA

AAAAAAAAAA

AAAAAAAAAA

AAAAAAAA


8.13 Eigenvectors and eigenvalues

xxxxx

Aμx xxA

Ax

xxAxA

vector normalized an is ,1 If )3(

ofr eigenvecto an is constant a is for )()( (2)

.eigenvalue ingcorrespond the

called is and , ofr eigenvecto an called is vector zero-non Any

vector. ldimensiona-N an is operatorlinear a is (1)

Ex: A non-singular matrix A has eigenvalues λi , and eigenvectors xi. Find the eigenvalues and eigenvectors of the inverse matrix A-1.

./1 are seigenvalue ingcorrespond thebut

, does as rseigenvecto same the has

/ 1

11

11

i

i

iiiii

i

ii

iii

i

AxA

xxAxAx

xAAxAxAx


Eigenvectors and eigenvalues of a normal matrix AAAA

. of seigenvalue the of conjugatecomplex the are of seigenvalue The

0)(

0)(0for (2)

matrix normal an also is

))((

))(()()(

00)(0set

0)( (1)

**

**

**

*

AA

xxAxIAxB

xBxBxBBxBxBx

B

BBIAIAAAAA

AAAA

IAIAIAIABB

BxBxBxBxBxIAB

xIAxAx


xxxx

xxxxxx

xxxAAx

xxAxx

xAxxAx

jijiji

jiji

jij

jii

ii

ii

ii

jij

ji

jij

jji

ii

0)( .orthogonal bemust and if

0))(()()(

)()()()( LHS

)()(

for and (3)

*

An eigenvalue corresponding to two or more different eigenvectors is said to be degenerate.

zxCAxCxCAAz

xCz

kixAx

ik

ii

ik

ii

ik

ii

ik

ii

kk

ii

11111

1

211

1

1

ncombinatiolinear

.......... , of any fromdifferent is

,....2,1for

degenerate fold-k is Suppose


productscalar s is )ˆ( (2)

])/[(ˆ ofvector unit the is ˆ (1)

ˆ])ˆ[(......ˆ])ˆ[(

ˆ])ˆ[(ˆ])ˆ[(

ˆ])ˆ[(

method. izationorthogonalSchmidt -Gram using by

orthogonal arethat rseigenvecto newConstruct

2/1

1111

13123233

12122

11

nm

iiiiii

kkkkkk

i

xz

zzzzzz

zxzzxzxz

zxzzxzxz

zxzxz

xz

z

yxaxay

Nixxx

ii

N

i

ii

jii

)( vector any

,...,3,2,1 ),)(( rseigenvecto lorthonormafor

1

ij


Ex: Show that a normal matrix can be written in terms of its

eigenvalues and orthogonal eigenvectors as

A

iix

)(

1

iiN

ii xxA

)(

)(

)( and vector arbitrary anfor

1

1111

1

iiN

ii

iiN

ii

ii

N

ii

iN

ii

iN

ii

ii

iN

ii

xxA

yxxxaAxaxaAAy

yxaxayy

Eigenvectors and eigenvalues of Hermitian and anti-Hermitian matrices

real are seigenvalue the )( Hermitian is if

),(matrix normal an is if *

AAA

xxAxAxAAAAA

real) is ( 0)(for ))((0

)()()()(for

)()()()(

**

**

*

iiiiiii

ii

iii

iii

iii

ii

ii

iii

iii

i

xxxx

xxxxxxAxxAA

xAxxAxxAx


Ex: Prove that the eigenvectors corresponding to different eigenvalues of an Hermitian matrix are orthogonal.

orthogonal 0)(for 0))(()2()1(

)2()()(

)1()()()(

)()()( and

seigenvalue twoFor

*

*

jiji

jiji

jij

ji

jii

jii

ji

ii

iijj

jii

i

ji

xxxx

xxAxx

xxxxxAx

xAxAxxAxxAx

anti-Hermitian matrix

zero.or imaginary pure are sEigenvalue

if

.orthogonal mutually are rseigenvecto The

normal also is )()(

*

*

xAxxAxxAx

AAAAAAAAAA


Unitary matrix

modulus.unit has Eigenvalue 1||

)( if

.orthogonal mutually are rseigenvecto The

normal also is

2*

*

111

xxAxAxAxAxxxxAx

AAAAIAAAAAA

Simultaneous eigenvectors

rs.eigenvecto ussimultaneo the have and

. ofr eigenvecto the also is

. eigenvlue to ingcorrespond ofr eigenvecto the is

)( commute and Suppose (1)

different. all are rsEigenvecto .....,3,2,1for

and matrices normal two are and if

BABAAB

BxxBx

ABx

BxBAxABx

BAABBA

NixAx

BBBBAAAANNBA

iiii

ii

ii

ii

ii

i


BAABBA

BAyBxAB

xcxcBxcBABAy

xcxcAxcABABy

xcy

xBxxAx

BA

iii

N

ii

iN

iii

N

i

ii

iN

iiii

ii

N

ii

iN

ii

iN

ii

ii

iii

i

rs.eigenvecto ussimultaneo have and

commute. and 0)(

vector arbitrary anyfor

and

common. in rseigenvecto the all have and Suppose (2)

111

111

1


8.14 Determination of eigenvalues and eigenvectors

0)()(

zero. is , oft deterninansecular the |,| .0||for

solution trivial-non a has only ),(0 If

equations. ussimultaneo

ofset shomogeneou a of form the has expression This

0)(matrix NN given A

1

N

iiTrA

ITrTrAIATr

AIABx

IAB Bx

xIAIxAxA

Ex: Find the eigenvalues and normalized eigenvectors of the real symmetric matrix

333

311

311

A


Sol:

0

2/1

2/1

2

110 ionnormalizat

0

and 0

2333

23

23

r eigenvecto

2for (1)

1 and 1

6 ,3 ,2

0)6)(3)(2(0

333

311

311

1221

213

3321

2321

13211

11

3

2

11

1

332211321

321

xkkkk

k

x

kxxx

xxxx

xxxx

xxxx

xAx

x

x

x

x

TrAAAA


xxxx

xx

xx-xxx

xxxx

xxxx

xxxx

xxxxxxx

xxx

xxxx

xxxx

xxxx

xxAx

x

x

x

x

TT 0)( and 0)( similarly,

0

1

1

1

3

1)011(

2

1)(:ityOrthogonal

2

1

1

6

12

6333

63

63

6for (3)

1

1

1

3

1

3

1

3

1

3

11 ionnormalizat

3333

33

33

3

3for )2(

3231

2T1

31312

3321

2321

1321

3

2321

23

22

21

321

3321

2321

132122

22

3

2

12

2


Degenerate eigenvalues

1

0

1

2

12for

0

1

0

2 (2)

1

0

1

2

1r eigenvecto

43

42

43

4 (1)

2 40)2)(4(0

-103

0-2-0

30-1

:

103

020

301

matrix thefor rseigenvecto ofset lorthonorma anConstruct :Ex

2133

22

1

331

22

121

1

3212

xxxx

x

xxx

xx

xxx

Sol

A


8.15 Change of basis and similarity transformations

XSXSXXxSx

eSxexexx

eSeNje

x

x

x

xxxxexxNie

N

jjiji

N

iiij

N

jjj

N

jji

N

ii

i

N

iijjj

N

TNi

N

iii

1''

1

'

11

''

1

'

1

1

''

2

1

211

formmatrix

,......2,1 , basis new afor

)...( formmatrix ......2,1 , basis A

ebasis

' ebasis

Sxvector

1S' xvector


Similarity transformations

and :vectorsfor

: basis in (2)

: basis in (1) :equationsOperator

1'

'1''

''

''''

ASSA

ASxSyASxAxSyy

SyySxx

xAye

Axye

i

i

similarity transformation

eSe 'for

Properties of the linear operator under two basis

TrAAA

ASSSASATrA

IAIASS

SIASIASSIA

ASSASASASSA

ISSISSAIAIA

jjjjk

j kkj

jkiji j k

kikijkiji j ki

ii

)()( :trace (4)

||||||||

|)(||||| :tdeterminan sticcharacteri (3)

|||||||||||||||| :tdeterminan (2)

(1)

11''

1

11'

111'

11''


If S is a unitary matrix:

ASSASSASS 1'1

IISS

ASASASSSASASSASSAAAA

AA

AASSSASASSAAA

AA

SSSSSSSS

SSeeSSeSeSee

e

ijijkjk

ikk

kjT

ikkjk

ki

krrjk r

kirkrjk r

kir

rrjk

kkiji

i

)()()(

unitary is unitary is if (3)

)()(

Hermitian)-(anti Hermitian is Hermitian)-(anti Hermitian is if (2)

)()(

lorthonorma is basis if (1)

''1

'

''

'

**

**''


8.16 Diagonalization of matrices

i

N

i

N

ii

N

ijjkjikk l

j

k

jkj

lik

jlkl

k likljkl

k likijij

jiij

N

jj

jj

i

N

iij

j

i

AATrATrAA

SS

xSxASSASASSA

xS

xxxxSS

xAxAx

ASSAeSx

NieA

1

'

1

'2

1

'

1

1111'

321

1'

1

|||| and

000

0...

0.0

0.0

)(

r.eigenvecto jth the ofcomponent ith the repreents

.

...

...

...

as expressed is and

),( of rseigenvecto the be to chosen are If

and by given is basis new A

,.....2,1 , basis a in drepresente is operator linear A


200

020

004

101

020

101

103

020

301

101

020

101

2

1

101

020

101

2

1 and

1

0

1

2

1

0

1

0

1

0

1

2

1

are rseigenvecto the example, previous the from :

103

020

301

matrix the eDiagonaliz :Ex

321

ASS

Sxxx

Sol

A

For normal matrices (Hermitian, anti-Hermitian and unitary) the N eigenvectors are linear independent.

ijjij

kk

ikkj

kkikj

kikij

jiij xxxxSSSSSSxS )()()( **


Ex: Prove the trace formula

0 !

expfor )exp(|exp|n

n

n

AATrAA

)exp()exp(exp|exp|

exp|)exp,.....exp ,exp ,(exp|

|)],...,(exp[||exp||exp|

|exp||exp||||||||exp||||exp|)(expexp

!!!

)(

))...()(()(

11

1321

2'

11'1'

0

1

0

1

0

'

1111'1'

TrAA

diag

diagAA

AASSSASASASA

Sn

ASS

n

AS

n

A

SASASSASSASSAASSA

N

iii

N

i

i

N

iN

Ni

n

nn

nn

n

nn


8.17 Quadratic and Hermitian forms

BxxAxxMxxQ

MMMB

MMMA

BAMMMMM

MxxQM

xxxxxxxxx

x

x

x

xxxMxxQ

MxxxQMxxxQ

TTT

T

T

TT

T

T

T

ofpart ricantisymmet the is )(2

1

ofpart symmetric the is )(2

1

)(2

1)(

2

1

and matrix, square real any is

6623

333

311

311

)( :Ex

)( formmatrix )(

32312123

22

21

3

2

1

321


. ofpart symmetric the only gconsiderin by unchanged is

)(0

)()(

quantity.scalar a isit ,For

MQ

AxxxBAxMxxBxx

BxxAxxBxxAxxQQ

BxxAxxxBxxAxBxxAxxQ

QQQ

TTTT

TTTTT

TTTTTTTTTTT

T

),......,,,(

terms-cross no )(.......)()()(

)()(

ed.diagonaliz be will , of rseigenvecto the by dconstructe is If

for )()(

tiontransforma Orthogonal

quantity.t independen-basis a is

321

2'2'33

2'22

2'11

'''''

'

''''''

1''

N

NN

TTT

TTTTT

T

diag

xxxx

xxxAxAxxQ

AAS

ASSAxAxASxSxAxxQ

xSxSxSxx

Q


real is )(6)(3)(2

6/)2(

3/)(

2/)(

211

222

033

6

1

and

220

123

123

6

1

2

1

1

6

1 ,6

1

1

1

3

1 ,3

0

1

1

2

1 ,2

333

311

311

)()()( form the into

6623

333

3-11

311

)(

form quadratic the takesthat of tiontransforma orthogonal an Find :Ex

2'3

2'2

2'1

321'3

321'2

21'1

3

2

1

'3

'2

'1

1'

33

22

11

2'33

2'22

2'11

32312123

22

21

3

2

1

321

QAxxxxxQ

xxxx

xxxx

xxx

x

x

x

x

x

x

xSxSxS

xxxA

xxx

xxxxxxxxx

x

x

x

xxxAxxQ

T

T

T


(real) )().(scalar a is

Hermitian is if )(

form Hermitian the ,expression general More

** HAxxxAxHHHHHH

AAAAxxxHTT

definite)-semi (posotive zero are seigenvalue of some 0

definite) (pisitive positive are of ' seigenvalue all0

ofr eigenvecto the basis the

as choosing by matrices column allfor ,

i

233

222

211

Q

AsQ

xxxQA

xAxxQ T


The stationary properties of the eigenvectors.

What are the vector that form a maximum or minimum of

xAxxxQ )(

xx

Axxx

QA

xxAxxQxAx

xxAxxAxx

AAxxxAxAxx

xxAxxAxx

xxAxx

T

T

TT

TT

TTTTT

TTT

TT

)( function the extremize to is procedure equivalent An(2)

s.eigenvalue ingcorrespond the to equal is value stationary the and

,stationary form quadratic the make of rseigenvecto The (1)

is solution the

0)()(

for ])()[()(

0)()(

0)]1([

smultiplier edundetermin Lagrange the Using


Ex: Show that if is stationary then is an eigenvector of and

is equal to the corresponding eigenvalues.

)(x x A )(x

points. stationary itsat )(

function the of values the are matrix symmetric a of seigenvalue The

.)( eigenvalue with ofr eigenvecto an is 00 if

][222

and )( using by

22

)]())([()(

1

in variations small torespect with 0)(

2

xx

Axxx

A

xλAxxAxxAx

xAxxxxxx

xx

xx

Axx

xxxxAxxxAxxAxxAx

xx

xx

xx

Axx

xx

Axx

xxxxAxxxAxAxxxxxx

xx

T

T

TTT

T

T

T

TTTTTTTT

T

T

T

T

T

T

TTTTTTT


8.18 Simultaneous linear equation

MNMNMM

N

N

M

M

MNMNMM

NN

NN

b

b

b

x

x

x

AAA

AAA

AAA

bAx

bbbb

bbbb

bxAxAxA

bxAxAxA

bxAxAxA

..

...

....

...

...

formMatrix

ousinhomogene called 0,....,, if

shomogeneou called 0,.....,, if

.....

........ .... .... ...

.......

.......

2

1

2

1

21

22221

11211

321

321

2211

22222121

11212111

bAxAA 10|| and square is if

:inversionDirect (1)

equationlinear ussimultaneo ofset sequare a Solve


4

3

2

7

0

4

8183

7134

212

11

1

7

0

4

233

221

342

:Sol

solutions. find and solution, unique a has

7233

022

4342

equations ussimultaneo ofset thethat Show :Ex

3

2

1

3

2

1

21

321

321

x

x

x

x

x

x

bAx

xxx

xxx

xxx


3

2

1

3

2

1

3323321331223212311131

2313212212211121

131211

2322

131211

3231

21

Obtain Obtain and Find

set

000

0

1

01

001

unity as ofelement diagonal the take we matrix, 33 aFor :Ex

matrix triangleupper square a :

matrix trianglelower square a :

iondecomposit LU (2)

x

x

x

x

y

y

y

yUL

bLyyUxbLUx

UULULULULUL

UULUULUL

UUU

UU

UUU

LL

LA

L

U

LLUA


4 ,3 ,2

2/11

2

4

8/1100

2/740

342

2/11 ,2 ,4

7

0

4

14/92/3

012/1

001

8/1100

2/740

342

14/92/3

012/1

001

8/112 :row 3rd

4/93:col. 2nd

2/7 42 2 :row 2nd

2/3 2/13 1 :col.1st 3 4 2 :row1st

7

0

4

233

221

342

equations

ussimultaneo ofset the solve to iondecomposit LU Use :Ex

321

3

2

1

321

3

2

1

333323321331

3222321231

2322231321221221

312111311121131211

3

2

1

xxx

x

x

x

yLx

yyy

y

y

y

bLy

LUA

UUULUL

LULUL

UUUULUUL

LLULULUUU

LUA

x

x

x


33231

22221

11211

3

33331

23221

13111

2

22323

23222

13121

1

33

22

1

1

33323

23222

13121

1

3332331

332

1

231

2322231

322

1

221

1312131

312

1

211

333231

232221

131211

3333232131

2323222121

1313212111

|| and

||onmanipulatiSimilar

1||

rule sCramer' (3)

bAA

bAA

bAA

AbA

AbA

AbA

AAb

AAb

AAb

Ax

Ax

xAAb

AAb

AAb

x

AAAx

xA

x

xA

AAAx

xA

x

xA

AAAx

xA

x

xA

AAA

AAA

AAA

A

bxAxAxA

bxAxAxA

bxAxAxA


Ex: Use Cramer’s rule to solve the set of the simultaneous equation.

7233

022

4342

321

321

321

xxx

xxx

xxx

411

44

|| 3

11

33

|| 2

11

22

||

44

733

021

442

33

273

201

342

22

237

220

344

11

233

221

342

||

33

22

11

321

Ax

Ax

Ax

A

8.1 vector spaces a set of vector is said to form a linear vector space v chapter 8 matrices and...

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antisymmetric matrices

functions of matrices

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set of vector