Positive Semidefinite matrix

Positive Semidefinite matrix nHA

nCzAzz 0*

A is a positive semidefinite matrix(also called nonnegative definite matrix)

Positive definite matrix

nHA

nCzAzz 0*

A is a positive definite matrix

Negative semidefinite matrix nHA

nCzAzz 0*

A is a negative semidefinite matrix

Negative definite matrix

nHA

nCzAzz 0*

A is a negative definite matrix

Positive semidefinite matrix

nT RxAxx 0

A is a positive semidefinite matrix

A is real symmetric matrix

Positive definite matrix

nT RxAxx 0

A is a positive definite matrix

A is real symmetric matrix

Question

nT RxAxx 0nCzAzz 0*

Yes

Is It true that)(RMALet n

?

Proof of Question

)(

)()()()(

))(()()(

)(

,,,

0)()(

***

*

yAxAxyiAyyAxxAxiyyAxiAyyAxx

yAxixAyiyAyxAxyAyyAixxAiyxAx

yiAxAiyxiyxAiyx

zAzAzziyxz

RyxwhereiyxzwritecanweCzanyForRxAxxthatAssume

clearisIt

TTTT

TTTT

TTTTTTTTTTTT

TTTTTTTT

TTTT

TTT

T

TT

n

n

nT

?

Proof of Question

)(

))(()()(*

AxyyAxiAyyAxxAxiyyAxiAyyAxxAyyyAixAxiyAxx

iAyxAiyxiyxAiyxAzz

TTTT

TTTT

TTTT

TT

TT

?

Fact 1.1.6 The eigenvalues of a Hermitian (resp. positive semidefinite , positive definite) matrix are all real (resp. nonnegative, positive)

Proof of Fact 1.1.6

edeifnegativeisAifesemideifnegativeisAif

edeifpositiveisAifesemideifpositiveisAif

andRzAzz

numberrealaisAzzAzzzAzAzzSince

zwherezAzz

zzzAzz

CzzAzthenreigenvectongcorresponithebez

andAofeigenvalueanbeLetHALet

n

n

int0int0

int0int0

,

.,)(

0,

0,,

.

2

*

*

*****

2

2

*

2**

Exercise n

n CzRAzzHA *

)(CMH n

nCzHzz 0*

From this exercise we can redefinite:

H is a positive semidefinite

注意 )(RMA n

nT RxAxx 0

A is symmetric

注意之反例 2

2

1

1

2

21

2

1

21 00110

R

01

10But is not

symmetric

Proof of Exercise

n

n

n

n

n

n

n

HAHenceAAAA

CzzAAzCzAzzzAzCzAzzAzzthen

CzRAzzthatAssumenumberrealaisAzz

thenAzzzAzAzz

CzanyForHAthatAssume

00

0)(

)(

)(.

,)(

.)(

*

*

**

***

***

*

*

*****

RemarkLet A be an nxn real matrix.

If λ is a real eigenvalue of A, then there must exist a corresponding real eigenvector.

However, if λ is a nonreal eigenvalue of A, then it cannot have a real eigenvector.

Explain of Remark p.1A, λ : real

Az= λz, 0≠z (A- λI)z=0 By Gauss method, we obtain that z is a real vector.

Explain of Remark p.2A: real, λ is non-real

Az= λz, 0≠z z is real, which is impossible

Elementary symmetric function

nnS 21211 ),,,(

21211

212 ),,,( iinii

nS

kiii

nkiiinkS

21

21121 ),,,(

kth elementary symmetric function

KxK Principal Minor

nxnijaALet niiianyFor k 211

kikiikiiki

kiiiiii

kiiiiii

aaa

aaaaaa

21

22212

12111

det

kxk principal minor of A

Lemma p.1nMALet

AofvectorcolumnithbeaLet i

niiianyFor k 211

kj

kj

j iiijifeiiijifa

bLet,,,,,,

21

21

vectordardsithbeeLet i tan

Lemma p.2 nbbbThen 21det

kiiibyindexedcolumnsand

rowswithorprincipalthe

,,,

min

21

Explain Lemma

4442

3432

2422

4442

3432

2422

1412

010

det

00100001

detaaaaaa

aaaaaaaa

4442

2422detaaaa

The Sum of KxK Principal Minors

AoforsprincipalkxkallofsumthebeAELet k

min)(

Theorem nMALet

AofpolynomialsticcharacterithebexcLet A )(

in

n

n

ii

in

A tSttcThen

),,,()1()( 211

AofseigenvaluethebeLet n ,,, 21

inn

ii

in tAEt

)()1(1

Proof of Theorem p.1

inn

ini

in

in

ijjj

n

i nijjj

n

nA

tSt

tt

ttttc

121

211 211

21

),,,()1(

)())((

)())(()()1(


LemmapreviousbytAEt

jjjkiftejjjkifa

bwhere

bbbt

ateateateAtItc

eeeILetAofvectorcolumnithisawhere

aaaALet

inn

ii

in

ik

ik

k

n

n

i nijjj

n

nn

A

n

i

n

,)()1(

,,,,,,

det

det)det()(

,)2(

1

21

21

211 211

2211

21

21

Rank P.1 rankA:=the maximun number of linear independent column vectors =the dimension of the column space = the maximun number of linear independent row vectors =the dimension of the row space

result

result

Rank P.2 rankA:=the number of nonzero rows in a row-echelon (or the reduced row echlon form of A)

Rank P.3

rankA:=the size of its largest nonvanishing minor (not necessary a principal minor) =the order of its largest nonsigular submatrix.

See next page

Rank P.4

0010

A

1x1 minorNot principal

minor

rankA=1

Theorem Let A be an nxn sigular matrix.Let s be the algebraic multiple of eigenvalue 0 of A.Then A has at least one nonsingular(nonzero)principal submatrix(minor) oforder n-s.


snorderoforprincipalnonzerooneleastathasA

AEtAEtAEt

formtheofistcsmultipleoftcofzeroaisSince

tAEttc

sn

s

sn

snnn

A

A

n

i

in

i

in

A

min

0)()()1()(

)(,)(0

)()1()(

1

1

1

Geometric multiple Let A be a square matrix and λ be aneigenvalue of A, then the geometric multiple of λ=dimN(λI-A) the eigenspace of A corresponding to λ

Diagonalizable

matrixdiagonalaisAPP

tsPgularnon

ifablediagonalizisA

1

.sin

Exercise A and have the same characteristic polynomial and moreover the geometric multiple and algebraic multiple are similarily invariants.

APP 1

Proof of Exercise p.1

)()det(

det)det(det))(det()det(

)det()()1(

1

1

11

1

1

xcAxI

PAxIPPAxIPAPPxPP

APPxIxc

A

APP

Proof of Exercise p.2 (2)Since A and have the same characteristic polynomial, they have the same eigenvalues and the algebraicmultiple of each eigenvalue is the same.

APP 1

Proof of Exercise p.3

)(dim)(dim)(dim)(dim

)(dim)(dim,,,,

)()(

)(,,2,1

)(,,,)(dim

)3(

1

1

1

21

1

1

21

1

1

APPEAEHenceAEAPPEhaveweSimilarly

APPEAEntindenpendelinearllyisPXPXPXSince

AEXPPXXPAXXAPP

riForAPPEforbasisabeXXXLet

APPErLetAPPandAofeigenvalueanbeLet

r

i

ii

ii

r

Explain: geom.mult=alge.mult in diagonal matrix

2lg325

))1,0,0,1,0((5))1,0,0,1,0((dim

))3,2,2,3,2()2,2,2,2,2((dim))3,2,2,3,2(2(dim2

32lg)3,2,2,3,2(2

ofmultipleebraicaThe

diagrankdiagN

diagdiagNdiagIN

ofmultiplegeometricTheofmultipleebraicaThediagofeigenvalueanis

Fact For a diagonalizable(square) matrix,the algebraic multiple and the geometric multiple of each of its eigenvalues areequal.

Corollary Let A be a diagonalizable(square) matrixand if r is the rank of A, then A has at least one nonsingular principalSubmatrix of order r.

Proof of Corollary p.1

rsnorderofsubmatrixprincipalnonsigularoneleastathasA

TheorempreviousBysn

ofmultipleebraicanofmultiplegeometricn

ANnrankArAofeigenvalueofmultiple

ebraicathebesLet

0lg0

)(dim0

lg

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Positive Semidefinite matrix