计算数学基础第2章

计算数学基础第2章沈复民

电子科技大学计算机科学与工程学院Chapter2UnitarySimilarityandUnitaryEquivalence2.0Introduction2.1UnitarymatricesandtheQRfactorization2.2Unitarysimilarity2.3Unitaryandrealorthogonaltriangularizations2.4ConsequencesofSchur’striangularizationtheorem2.5Normalmatrices2.6Unitaryequivalenceandthesingularvalueposition2.7TheCSpositionDefinition1.3.1Let

A,B

Mnbegiven.Wesaythat

BissimilartoA

ifthereexistsanonsingularmatrixS

MnsuchthatB=S1ASThetransformationA

S

1A

SiscalledasimilaritytransformationbythesimilaritymatrixS.Therelation“BissimilartoA”issometimesabbreviatedB

A.1.3SimilarityS1=S

2.0IntroductionSimilarityviaaunitarymatrixU,A

U

AU,isnotonlyconceptuallysimplerthangeneralsimilarity(theconjugatetransposeismucheasiertocomputethantheinverse),butitalsohassuperiorstabilitypropertiesinnumericalcomputations.2.0IntroductionForA

Mn,m,thetransformationA

UAV,inwhichU

MmandV

Mnarebothunitary,iscalledunitaryequivalence.ThetransformationA

S

AS,inwhichS

is

nonsingularbutnot

necessarilyunitary,iscalled

congruence;westudyitinChapter4.Definition2.1.12.1UnitarymatricesandtheQRfactorizationAlistofvectorsx1,,xkCnisorthogonalifforalli

j

,i,j{1,,k}.If,inaddition,forall

i=1,,k(thatis,thevectorsarenormalized),

thenthelistisorthonormal.Theorem2.1.2EveryorthonormallistofvectorsinCnislinearlyindependent.2.1UnitarymatricesandtheQRfactorizationAlinearlyindependentlistneednotbeorthonormal,ofcourse,butonecanapplytheGram-Schmidtorthonormalizationprocedure

(0.6.4)

toitandobtainanorthonormallistwiththesamespan.2.1UnitarymatricesandtheQRfactorizationDefinition2.1.3U

=U1

AmatrixU

Mn

(C)isunitary

if

U

U=I.AmatrixU

Mn(R)isrealorthogonalif

UTU=I.DefinitionAmatrixA

MnissaidtobeHermitian

if

A

=A.If,inaddition,A

Mn(R),Aissaidtobesymmetric.AmatrixA

Mnissaidtobenormal

if

A

A=AA

,thatis,if

AcommuteswithitsHermitianadjoint.AmatrixU

Mnissaidtobeunitary

if

U

U=I.If,inaddition,U

Mn(R),Uissaidtoberealorthogonal.7.1DefinitionsandpropertiesAHermitianmatrixA

Mnispositivedefinite

ifx

Ax0forallnonzeroxCn.(7.1.1a)Itispositivesemidefinite

ifx

Ax0forallnonzeroxCn.(7.1.1b)Implicitinthesedefinitionsisthefactthatif

AisHermitian,then

x

AxisrealforallxCn;see(4.1.3).Conversely,if

A

Mnandx

AxisrealforallxCn,then

AisHermitian,soassumingthatAisHermitianintheprecedingdefinitions,whilecustomary,isactuallysuperfluous;see(4.1.4).Theorem7.2.17.2CharacterizationsandpropertiesAHermitianmatrixispositivesemidefinite

ifandonlyifallofitseigenvaluesarenonnegative.Itispositivedefinite

ifandonlyifallofitseigenvaluesarepositive.SeveralClassesofMatricesChapters2,4,and7HermitianmatrixChapters4Positive(semi)definitematrixChapters7NormalmatrixChapters2UnitarymatrixChapters2Theunitarymatricesin

Mnformaremarkableandimportantset.WelistsomeofthebasicequivalentconditionsforUtobeunitaryin(2.1.4).7.2CharacterizationsandpropertiesTheorem2.1.4

IfU

Mn,thefollowingareequivalent:

(a)

Uisunitary.(U

U=I)

(b)

UisnonsingularandU

=U

1.

(c)

UU

=I.

(d)

U

isunitary.

(e)ThecolumnsofUareorthonormal.

(f)TherowsofUareorthonormal.

(g)ForallxCn,||x||2=||Ux||2,thatis,xandUxhavethesameEuclideannorm.Definition2.1.5AlineartransformationT:

CnCmiscalledaEuclideanisometry

if

||x||2=||Tx||2

forallxCn.Theorem2.1.4saysthatasquarecomplexmatrixU

MnisaEuclideanisometry(viaU:x

Ux)ifandonlyifitisunitary.2.1UnitarymatricesandtheQRfactorizationObservation2.1.6

If

U,V

Mnareunitary(respectively,realorthogonal),thentheUVisalsounitary(respectively,realorthogonal).Theorem2.1.4(b)2.1UnitarymatricesandtheQRfactorizationObservation2.1.7Thesetofunitary(respectively,realorthogonal)matricesinMn

formsagroup

.Thisgroupisgenerallyreferredtoasthen-by-nunitary(respectively,realorthogonal)group,asubgroupofGL(n,C)

(0.5).2.1UnitarymatricesandtheQRfactorization0.5NonsingularityThenonsingularmatricesinMn(F)

formagroup,thegenerallineargroup,oftendenotedbyGL(n,F)

GL(n,C)F=CNotionsof“convergence”and“limit”ofasequenceofmatricesarepresentedprecisely

inChapter5.2.1UnitarymatricesandtheQRfactorizationLemma2.1.8(TheSelectionPrincipleforUnitaryMatrices)2.1UnitarymatricesandtheQRfactorizationLet

U1,U2,Mnbeagiveninfinitesequenceofunitarymatrices.ThereexistsaninfinitesubsequenceUk1,Uk2,,1k1<k2<,suchthatalloftheentriesof

Ukiconverge(assequencesofcomplexnumbers)totheentriesofaunitarymatrixasi

.ThesetofsuchmatricesiseasilycharacterizedastherangeofthemappingA

A1A

forallnonsingularA

Mn.AunitarymatrixUhasthepropertythatU

1

equals

U

.OnewaytogeneralizethenotionofaunitarymatrixistorequirethatU

1besimilartoU

.2.1UnitarymatricesandtheQRfactorizationTheorem2.1.9Let

A

Mnbenonsingular.Then

A1issimilartoA

ifandonlyifthereisanonsingularmatrixB

MnsuchthatA=B1B

.A{B1B

,B

Mnisanonsingularmatrix}A

1issimilartoA

2.1UnitarymatricesandtheQRfactorization2.1UnitarymatricesandtheQRfactorizationIfaunitarymatrixispresentedasa2-by-2blockmatrix,thentheranksofitsoff-diagonalblocksareequal

;theranksofitsdiagonalblocksarerelatedbyasimpleformula.Lemma2.1.102.1UnitarymatricesandtheQRfactorizationLetaunitarymatrixU

Mnbepartitionedas,inwhichU11

Mk.Then

rankU12=rankU21andrankU22=rankU11+n2k.Inparticular,U12=0

ifandonlyifU21=0,inwhichcaseU11andU22

areunitary.

PlanerotationsandHouseholdermatricesarespecial(andverysimple)unitarymatricesthatplayanimportantroleinestablishingsomebasicmatrixfactorizations.2.1UnitarymatricesandtheQRfactorizationcolumnicolumnjrowirowjdenotetheresultofreplacethe

i,iand

j,j

entriesofthen-by-nidentitymatrixbycos

,replacingits

i,j

entry

by

sin

andreplacingits

j,i

entryby

sin

.ThematrixU(

;i,j

)iscalledaplanerotationorGivensrotation.Example2.1.11.Planerotations.Let1

i<j

n

andletU(

;i,j

)Exercise2.1UnitarymatricesandtheQRfactorizationVerifythat

U(

;i,j)

Mn(R)isrealorthogonalforanypairofindicesi,jwith1i<j

nandanyparameter

[0,2

).ThematrixU(

;i,j)carriesoutarotation(throughanangle

)inthei,jcoordinateplaneofRn.2.1UnitarymatricesandtheQRfactorizationExample2.1.12.HouseholdermatricesLetw

Cnbeanonzerovector.TheHouseholdermatrixUw

MnisdefinedbyUw=I2(w

w)1ww

.If

wisaunitvector,then

Uw=I2ww

.Exercise2.1UnitarymatricesandtheQRfactorizationShowthataHouseholdermatrixUw

isboth

unitary

andHermitian,soUw1=Uw.ShowthataHouseholdermatrixUw

actsastheidentityonthecomplementarysubspacew

andthatitactsasareflectionontheone-dimensionalsubspacespannedbyw;thatis,Uw

x=x

if

x

w

and

Uw

w=w.2.1UnitarymatricesandtheQRfactorizationExerciseThefollowingQRfactorizationofacomplexorrealmatrixisofconsiderabletheoreticalandcomputationalimportance.2.1UnitarymatricesandtheQRfactorization(a)

If

n

m,thereisaQ

Mn,mwithorthonormalcolumnsandanuppertriangularR

MmwithnonnegativemaindiagonalentriessuchthatA=QR.

(b)

If

rankA=m,thenthefactorsQandRin(a)areuniquelydeterminedandthemaindiagonalentriesofRarepositive.

(c)

If

m=n,thenthefactorQin(a)isunitary.

(d)ThereisaunitaryQ

Mnandanuppertriangular

R

Mn,m

withnonnegativediagonalentriessuchthatA=QR.

(e)

If

Aisreal,thenthefactorsQandRin(a),(b),(c),and(d)

maybetakentobereal.Theorem2.1.14(QRfactorization)

LetA

Mn,mbegiven.QRfactorizationandQRalgorithmApopularnumericalmethodfor

calculatingeigenvalues(undersomeassumptions)iscalledtheQRalgorithm.ItisbasedontheQRfactorization.QRalgorithm

Let

A0

Mn

begiven.WriteA0=Q0R0,whereQ0andR0areasguaranteedinTheorem2.1.14,anddefineA1=R0Q0.Again,writeA1=Q1R1,withQ1unitaryandR1uppertriangular,andcontinue.Ingeneral,factorAk=Qk

RkanddefineAk+1=Rk

Qk

.QRfactorizationandQRalgorithmExercise

ShowthateachAkproducedbytheQRalgorithmisunitarilysimilartoA0,k=1,2,.QkRk

QkAk

Qk=QkAk+1=Ak+1=Qk

AkQkQkAk+1=AkQk

Undercertaincircumstances(forexample,ifalltheeigenvaluesofA0havedistinct

absolutevalues),theQRiteratesAkwillconvergetoanuppertriangularmatrixask

.SincethisuppertriangularmatrixisunitarilyequivalenttoA0,theeigenvaluesofA0arerevealed.QRfactorizationandQRalgorithmAnimportantgeometricalfactisthatanytwolistscontainingequalnumbersoforthonormalvectorsarerelatedviaaunitarytransformation.2.1UnitarymatricesandtheQRfactorizationTheorem2.1.182.1UnitarymatricesandtheQRfactorizationIf

X=[x1

xk]

Mn,kandY=[y1

yk]Mn,khaveorthonormalcolumns,thenthereisaunitaryU

MnsuchthatY=UX.If

XandYarereal,then

Umaybetakentobereal.Problems2.1UnitarymatricesandtheQRfactorization1.

IfU

Mnisunitary,showthat

|detU|=1.2.

Let

U

Mnbeunitaryandlet

beagiveneigenvalueofU.

Showthat

(a)

|

|=1and(b)

xisa(right)eigenvectorofU

associatedwith

ifandonlyif

xisalefteigenvectorofU

associatedwith

.9.

If

U

Mnisunitary,showthat

U,U

T,andU

areallunitary.–12.

Showthatif

A

Mnissimilartoaunitarymatrix,then

A

1issimilartoA

.2.2UnitarysimilaritySince

U

=

U1foraunitary

U

,thetransformationonMngivenbyA

U

AU

isasimilaritytransformationifUisunitary.Thisspecialtypeofsimilarityiscalledunitarysimilarity.Definition2.2.12.2UnitarysimilarityLet

A,B

Mnbegiven.WesaythatAisunitarilysimilartoBifthereisaunitaryU

MnsuchthatA=UBU

.If

Umaybetakentobereal(andhenceisrealorthogonal),then

Aissaidtobe(real)

orthogonallysimilartoB.WesaythatAisunitarilydiagonalizable

ifitisunitarilysimilartoadiagonalmatrix;Aisrealorthogonallydiagonalizable

ifitisrealorthogonallysimilartoadiagonalmatrix.ExerciseShowthatunitarysimilarityisanequivalencerelation.2.2UnitarysimilarityLet

A,

B

Mn.If

BissimilartoA,then

AandBhavethesamecharacteristicpolynomial.Theorem1.3.31.3SimilarityLet

A,

B

MnandsupposethatAissimilartoB,then(a)

AandBhavethesameeigenvalues.

(b)IfBisadiagonalmatrix,itsmaindiagonalentriesaretheeigenvaluesofA.

(c)

B=0(adiagonalmatrix)ifandonlyifA=0.

(d)

B=I(adiagonalmatrix)ifandonlyifA=I.Corollary1.3.41.3SimilarityThedeterminant,trace,andrankaresimilarityinvariants.rankS1AS=rankAtrS1AS=trAdetS1AS=detATheorem2.2.22.2UnitarysimilarityLet

U

MnandV

Mmbeunitary,let

A=[aij]Mn,m

and

B=[bij]

Mn,m,andsupposethat

A=UBV.Then.Inparticular,thisidentityissatisfiedif

m=nandV=U

,thatis,if

AisunitarilysimilartoB.trB

B

=trA

ATheorem2.2.2saysthattrA

Aisunitarysimilarityinvariant.ExerciseShowthatthematrices

andaresimilarbutnotunitarilysimilar.2.2UnitarysimilarityDefinition1.3.1Let

A,B

Mnbegiven.Wesaythat

BissimilartoA

ifthereexistsanonsingularmatrixS

MnsuchthatB=S1ASThetransformationA

S

1A

SiscalledasimilaritytransformationbythesimilaritymatrixS.Therelation“BissimilartoA”issometimesabbreviatedB

A.1.3SimilarityS1=S

2.2UnitarysimilarityUnitarysimilarity,likesimilarity,correspondstoachangeofbasis,butofaspecialtype–itcorrespondstoachangefromoneorthonormal

basistoanother.Unitarysimilarityimpliessimilaritybutnotconversely.TheunitarysimilarityequivalencerelationpartitionsMnintofinerequivalenceclassesthanthesimilarityequivalencerelation.Theorem2.2.22.2UnitarysimilarityLet

U

MnandV

Mmbeunitary,let

A=[aij]Mn,m

and

B=[bij]

Mn,m,andsupposethat

A=UBV.Then.Inparticular,thisidentityissatisfiedif

m=nandV=U

,thatis,if

AisunitarilysimilartoB.trB

B

=trA

AChapter2UnitarySimilarityandUnitaryEquivalence2.0Introduction2.1UnitarymatricesandtheQRfactorization2.2Unitarysimilarity2.3Unitaryandrealorthogonaltriangularizations2.4ConsequencesofSchur’striangularizationtheorem2.5Normalmatrices2.6Unitaryequivalenceandthesingularvalueposition2.7TheCSposition2.3UnitaryandrealorthogonaltriangularizationsPerhapsthemostfundamentallyusefulfactofelementarymatrixtheoryisatheoremattributedtoIssaiSchur:AnysquarecomplexmatrixAisunitarilysimilartoatriangularmatrixwhosediagonalentriesaretheeigenvaluesofA,inanyprescribedorder.Theorem2.3.1(Schurform:Schurtriangularization)Let

A

Mnhaveeigenvalues

1,,

ninanyprescribedorderandlet

xCnbeaunitvectorsuchthatAx=

1x.(a)ThereisaunitaryU={x,u2

un}MnsuchthatU

AU=T=[ti

j]isuppertriangularwithdiagonalentriesti

i=

i

,

i=1,,n.(b)

IfA

Mn(R)hasonlyrealeigenvalues,then

xmaybechosentoberealandthereisarealorthogonalQ={x

q2

qn}Mn(R)

suchthatQTAQ=T=[tij]isuppertriangularwithdiagonalentriestii=

i,i=1,,n.2.3UnitaryandrealorthogonaltriangularizationsNotethat

neithertheunitarymatrixU

northeuppertriangularmatrixTofTheorem2.3.1isunique.NotonlymaythediagonalentriesofT(theeigenvaluesofA)appearinanyorder,butalsotheunitarilysimilaruppertriangularmatricesmayappearverydifferentabovethediagonal.Example2.3.2VerifythatU

isunitaryand

T2=UT1U

.,.,IftheeignevaluesofAarereorderedandthecorrespondinguppertriangularization(2.3.1)isperformed,theentriesofTabovethemaindiagonalcanbedifferent.Consider2.3UnitaryandrealorthogonaltriangularizationsThereisausefulextensionof(2.3.1):Acommutingfamilyofcomplexmatricescanbereducedsimultaneouslytouppertriangularformbyasingleunitarysimilarity.Theorem2.3.32.3UnitaryandrealorthogonaltriangularizationsLet

F

Mnbeanonemptycommutingfamily.ThereisaunitaryU

MnsuchthatU

AUisuppertriangularforevery

A

F.2.3UnitaryandrealorthogonaltriangularizationsIfarealmatrixAhasanynon-realeigenvalues,thereisnohopeofreducingittoauppertriangularform

TbyarealsimilaritybecausesomemaindiagonalentriesofT(eigenvaluesofA)wouldbenon-real.However,wecanalwaysreduceAtoarealupperquasitriangularform

Tbyarealorthogonalsimilarity;conjugatepairsofnon-realeigenvaulesareassociatedwith2-by-2blocks.0.9.4BlocktriangularmatricesAblockuppertriangularmatrixinwhichallthediagonalblocksare1-by-1or2-by-2issaidtobeupperquasitriangular.Amatrixislowerquasitriangular

ifitstransposeisupperquasitriangular;itisquasitriangular

ifitiseitherupperquasitriangularorlowerquasitriangular.Amatrixthatisbothupperquasitriangularandlowerquasitriangularissaidtobequasidiagonal.withthefollowingproperties:(i)its1-by-1diagonalblocksdisplaytherealeignevaluesofA;(ii)eachofits2-by-2diagonalblockshasaspecialformthatdisplaysaconjugatepairofnon-realeigenvaluesofA:(iii)itsdiagonalblocksarecompletelydeterminedbytheeigenvaluesofA

;theymaybeappearinanyprescribedorder.Theorem2.3.4(RealSchurform)

Let

A

Mn(R)begiven.,a,b

R,b>0,anda

ibareeigenvaluesof

A.(2.3.5a)(a)Thereisarealnonsingular

S

Mn(R)suchthatS1ASisarealupperquasitriangularmatrix,eachAiis1-by-1or2-by-2(2.3.5)(i)Its1-by-1diagonalblocksdisplaytherealeignevaluesofA;(ii)Eachofits2-by-2diagonalblockshasaconjugatepairofnon-realei-genvalues(butnospecialform);(iii)Theorderingofitsdiagonalblocksmaybeprescribedinthefollowingsense:Iftherealeigenvaluesandconjugatepairsofnon-realeigenvluesofA

arelistedinaprescribedorder,thentherealeigenvaluesandconjugatepairsofnon-realeigenvaluesoftherespectivediagonalblocksA1,,AmofQTAQareinthesameorder.Theorem2.3.4(RealSchurform)

Let

A

Mn(R)begiven.(b)Thereisarealorthogonal

Q

Mn(R)suchthatQ

TAQisarealupperquasitriangularmatrixwiththefollowingproperties:2.3UnitaryandrealorthogonaltriangularizationsThereisacommutingfamiliesversionof

Theorem2.3.4:Acommutingfamilyofrealmatricesmaybereducedsimultaneouslytoacommonupperquasitriangularformbyasinglerealorthogonalsimilarity.Problem6.

Let

A,

B

Mnbegiven,andsuppose

AandBaresimultaneouslysimilartouppertriangularmatrices;thatis,S

1ASandS

1

BS

arebothuppertriangularforsomenonsingularS

Mn.ShowthateveryeigenvaluesofAB

BAmustbezero.2.3Unitaryandrealorthogonaltriangularizations2.4ConsequencesofSchur’striangularizationtheoremUse(2.3.1)toshowthatif

A

Mnhaseigenvalues

1,,

n,countingmultiplicity,then

detA=andtrA=.2.4.1ThetraceanddeterminantTheorem2.3.1(Schurform:Schurtriangularization)Let

A

Mnhaveeigenvalues

1,,

ninanyprescribedorderandlet

xCnbeaunitvectorsuchthatAx=

1x.(a)ThereisaunitaryU={x,u2

un}MnsuchthatU

AU=T=[ti

j]isuppertriangularwithdiagonalentriesti

i=

i

,

i=1,,n.(b)

IfA

Mn(R)hasonlyrealeigenvalues,then

xmaybechosentoberealandthereisarealorthogonalQ={x

q2

qn}Mn(R)

suchthatQTAQ=T=[tij]isuppertriangularwithdiagonalentriestii=

i,i=1,,n.2.4ConsequencesofSchur’striangularizationtheorem2.4.3TheCaley-Hamiltontheorem

Thefactthateverysquarecomplexmatrixsatisfiesitsowncharacteristicequationfollowsfrom

Schur’stheoremandanobservationaboutmultiplicationoftriangularmatriceswithspecialpatternsofzeroentries.Lemma2.4.3.1

Supposethat

R=[rij]and

T=[tij]Mnareuppertriangularandthatri

j=0

,1i,j

k

n,andt

k+1,k+1=0.Let

S==RT.Then

,

1i,j

k+1.2.4ConsequencesofSchur’striangularizationtheoremkkk+1k+1k+1k+12.4ConsequencesofSchur’striangularizationtheoremLemma2.4.3.1Theorem2.4.3.2(Cayley-Hamilton)Let

pA

(t)bethecharacteristicpolynomialofA

Mn

.Thenp

A(A)=02.4ConsequencesofSchur’striangularizationtheoremExerciseP.58Example1.3.5

&P.118Example2.4.8.42.4ConsequencesofSchur’striangularizationtheoremWhatiswrongwiththefollowingargument?“SincepA(

i)=0foreveryeigenvalue

iofA

Mn,andsincetheeigenvaluesofpA

(A)arepA(

1),,pA(

n),alleigenvaluesofpA(A)are0.Therefore,pA(A)

=0.”Itisacommonmistakenargumentfor

theCayley-Hamiltontheorem.Givenanexampletoillustratethefallacyintheargument.ExerciseThescalarpolynomialpA(t)isfirstcomputedaspA(t)=det(t

I

A),andonethenformsthematrixpA(A)fromthecharacteristicpolynomial.2.4ConsequencesofSchur’striangularizationtheoremWhatiswrongwiththefollowingargument?“SincepA(t)=det(t

I

A),wehavepA(A)=det(AI

A)=det(A

A)=det0=0.Therefore,pA(A)=

0.”

Theorem2.4.3.2(Cayley-Hamilton)Let

pA

(t)bethecharacteristicpolynomialofA

Mn

.Thenp

A(A)=02.4ConsequencesofSchur’striangularizationtheoremTheCayley-Hamiltontheoremisoftenparaphrasedas“

everysquarematrixsatisfiesitsowncharacteristicequation(1.2.3)”,butthismustbeunderstoodcarefully:ThescalarpolynomialpA(t)isfirstcomputedaspA(t)=det(tI

A);onethencomputesthematrixpA(A)bysubstitutingt

A.2.4ConsequencesofSchur’striangularizationtheoremTheorem2.4.3.2(Cayley-Hamilton)OneimportantuseoftheCayley-HamiltontheoremistowritepowersAkofA

Mn,fork

n,aslinearcombinationsofI,A,A2,,An1.2.4ConsequencesofSchur’striangularizationtheoremTheorem2.4.3.2(Cayley-Hamilton)Example2.4.3.3LetThen

pA(t)=t2

3t+2,so

A2

3A+2I=0.Thus,

A2=3A

2I;A3=A(A2)=3A2

2A=3(3A

2I)

2A=7A

6I;A4=7A2

6A=15A

14I,andsoon.

WecanalsoexpressnegativepowersofthenonsingularmatrixA

aslinearcombinationsofAandI.WriteA2

3A+2I=0

as2I=

A2+3A=A(

A+3I),or

I=A[(

A+3I)].Thus,A

1=

A+I=

,A

2=(A+I)2=A2

A+I=(3A2I)A+I

=A+I

,andsoon.Corollary2.4.3.4Suppose

A

Mn

isnonsingularandlet

pA(t)=tn+an1

t

n1++a1

t+a0.Let

q(t)=(tn1+an1

tn2++a2

t+a1)/a0.

Then

A

1=q(A)isapolynomialinA.2.4ConsequencesofSchur’striangularizationtheoremGivensomethoughttotheconverse:Satisfactionofthesamepolynomialequationsimpliessimilarity-trueorfalse?Exercise2.4ConsequencesofSchur’striangularizationtheoremIf

A,B

Mnaresimilarandg(t)isanygivenpolynomial,showthat

g(A)issimilartog(B),andthatanypolynomialequationsatisfiedbyAissatisfiedbyB.WehaveshownthateachA

Mnsatisfiesapolynomialequationofdegreen,forexample,itscharacteristicequation.ItispossibleforA

Mntosatisfyapolynomialequationofdegreelessthan

n,however.2.4ConsequencesofSchur’striangularizationtheoremExample2.4.3.5Example2.4.3.5ConsiderThecharacteristicpolynomialispA(t)=(t1)3andindeed(A

I)3=0.But(A

I)2=0soAsatisfiesapolynomialequationofdegree2.2.4ConsequencesofSchur’striangularizationtheoremTheequation

AXXA=0

associatedwithcommutativityisaspecialcaseofthelinearmatrixequationAX

XB=C,oftencalledSylvester’sequation.Thefollowingtheoremgivesanecessaryandsufficientconditionfor

Sylvester’sequationtohaveauniquesolutionX