Lecture3MatrixNormsandMatrixEigenvalues

1Lecture3:MatrixNormsandMatrixEigenvaluesMATRIXANALYSIS@HITSZINSTRUCTOR:You-HuaFanReadingassignmentonthetextbook•Section5.6.1~5.6.6(Page290-296)•Section1.1,1.2,1.3.1~1.3.12,1.3.20,1.4.1~1.4.4(Page34-58)•Section5.6.7~5.6.16,5.8(Page296-301,335-339)矩阵范数矩阵的特征值2Inthislecturewebeginourstudyofthemostimportantaspectofmatrixtheory.Calledspectraltheory,itallowsustogivefundamentalstructuretheoremsformatricesandtodeveloppowertoolsforcomputing.Webeginwithastudyofnormsonmatrices.WeknowMncanbeseenasan2dimensionalvectorspace,butithasaadditionaloperation,matrixproduct.Howtodefineametricsforamatrixconsideringthematrixproductoperation?3.1BasicConceptsonMatrixNorms谱理论3Definition3.1.1.Wecallafunction|||·|||:MnR+amatrixnormifithasthefollowingproperties:icativeSubmultipl||||||||||||||||||)4(inequalityTriangle||||||||||||||||||)3(sHomogeneouallfor||||||||||||||)2(Positive0iff0||||||and0||||||)1(BAABBABACcAccAAAAQuestions:(a)Ismatrixnormacontinuousfunction?(b)Areallmatrixnormsequivalent?矩阵范数正定性齐次性三角不等式次乘性4Exercise.For,proof:nMA.1||||||)(||||||||||||||||||then,invertibleisif)(1||||||thenif)(,...3,2,||||||||||||,||||||||||||)(1222IdAIAAcAAAbpAAAAapp5Example.Thel1normisamatrixnorm.ProofWejustverifythesubmultiplicative..||||||||||||||||||||||||||||11,,,,,,,,1BAbababababaABnjnjmiimnmjinjimjinmnjimjikkjikjikkjikl1范数6Example.Thel2normisamatrixnorm(Frobeniusnorm)ProofWejustverifythesubmultiplicative.UsingCauchy-Schwarzinequality,wehave:.||||||||||||||||||||2222,2,22,22,22BAbababaABjnnjmiimnnjjimimjikkjikl2范数F范数7Example.Thel∞normisnotamatrixnorm.ProofConsiderthematrix:icative.submultiplthesatisfytdoesn'itso,||||||||12||||2||||.2,1||||1111222JJJJJJJJExercise.Proofisamatrixnorm,wherenisthesizeofthematrix.||||nl无穷范数8Definition3.1.2.Let||·||beavectornormonCn.Definethefunction|||·|||onMnby||||||||max||||max||||||0||||1||||xAxAxAxxanditiscalledthematrixnormdeducedbythevectornorm(oroperatornorm)Exercise.Fortheoperatornorm|||·|||:(a)Verifyitsthetriangularinequalityandsubmultiplicative.(b)Forall,proof(c)Proof|||I|||=1.(d)IfAisinvertible,proof.||||||||||||||xAAxnMA.|||)(|||||||||11AA算子范数9Briefproof.||||||||||||||||max||||max||||||||max||)(||max||||||)(BABxAxBxAxxBABAa||||||||||||||||||||max||||||||max||||||||||||||||max||||||||max||||||BAxBxyAyxBxBxABxxABxAB0||||||||||||||,0if||||||||||||||||,||/||||||||||,0if)(xAAxxxAAxxAxAxb11|||)(|||||||||/||||||||||||1||||/||||max||||||)()(AAIAxIxIdc10Example.Themaximumcolumnsumnorm|||·|||1isdeducedbythel1norm.iijjaA||max||||||1Proof.1111||||||||||||||||||||||max||||||||||||xAxAxaxaxaxaAxjjjjiijjjjiijijjijijjijiiliijjaaxAxA||||maxlet.||||/||||||||||1111111||||/|||||||||)(|||||||||llliiliileAeAeAeaA110||||1||||/||||max||||||xAxAx最大列和范数11Example.Themaximumrowsumnorm|||·|||∞isdeducedbythel∞norm.jijiaA||max||||||Proof.||||||||||||max||max||||maxmax||||xAxaxaxaAxjjjijijjijijjijijljjijiaaxAxA||||maxlet.||||/||||||||||||||/|||||||||)(|max|)(|||||||||zAzAzAzAzeaeaAkkljiljjiljjj最大行和范数12Example*.Thespectralnorm|||·|||2isdeducedbythel2norm.AAofeigenvalueanisA:max||||||2Proof.2220222||||maxmax||||max()||||||xjjAxxAAxxxxAAA谱范数特征值133.2EigenvaluesandEigenvectorsExample.ConsiderthematrixWehave1427A2136321142)2(1721142721A1155511141)2(1711142711A•Eigenvalues特征向量特征值14Note:Theeigenvalueandeigenvectoroccurinextricablyasapair,andaneigenvectorcannotbethezerovector.Definition3.2.1.Ifand,weconsidertheequation(eigenvalue-eigenvectorequation)whereisascalar.Ifascalarandanonzerovectorhappentosatisfythisequation,theniscalledaneigenvalueofAandiscalledaneigenvectorofApertainingto.nMAnxC0,xxAxxx特征方程15Exercise.Considertheoptimizationproblemofmaximizingarealsymmetricquadraticformsubjectedtoageometricconstraint:Maximize,subjecttoAxxT1,xxxTnRinwhichisgiven.)(RnTMAA16xxxxAxxTTexT)(|Thatis,theextremumistheeigenvalueofA.Themaximumisthelargesteigenvalue.andistheengenvectorpertainingto.xmaxmaxSolution.IntroducetheLagrangian.Necessaryconditionsforanextremumare:xxAxxLTT0)(20xAxLThus,theextremumofmustrequire,thatis,isaneigenvectorofA.Andtheextremumis:AxxTxxAx17Definition3.2.2.ThesetofalleigenvaluesofamatrixAiscalledthespectrumofAandisdenotedbyDefinition3.2.3.ThespectralradiusofAisthemaximummodulusofitseigenvaluesandisdenotedby.vectornonzeroaforsolutionahas|)(xxAxA)}.(|:max{|)(AANote.ThespectralradiusisjusttheradiusofthesmallestdisccenteredattheorigininthecomplexplanethatincludesalltheeigenvaluesofA.(seeFig3.2.1)谱谱半径18O)(A)Im()Re(Fig3.2.1Smallestdiscincludesalltheeigenvalues19Theorem3.2.1.Letp(·)beagivenpolynomial.Ifisaneigenvalueof,andthepertainingeigenvectoris,thenisaneigenvalueofthematrixandisapertainingeigenvector.ThatisnMAx)(p)(ApProof:xaAxaxAaxAaxApkkkk0111)(let