老师布置作业第二章

EigenvaluesandE=EIG(X)isavectorcontainingtheeigenvaluesofasquarematrixX.[V,D]=EIG(X)producesadiagonalmatrixDofeigenvaluesandafullmatrixVwhosecolumnsarethecorrespondingeigenvectorssothatX*V=V*D.[V,D]=EIG(X,'nobalance')performsthecomputationwithbalancingdisabled,whichsometimesgivesmoreaccurateresultsforcertainproblemswithunusualscaling.IfXissymmetric,EIG(X,'nobalance')isignoredsinceXisalreadybalanced.E=EIG(A,B)isavectorcontainingthegeneralizedeigenvaluesofsquarematricesAandB.[V,D]=EIG(A,B)producesadiagonalmatrixDofgeneralizedeigenvaluesandafullmatrixVwhosecolumnsarethecorrespondingeigenvectorssothatA*V=B*V*D.EIG(A,B,'chol')isthesameasEIG(A,B)forsymmetricAandsymmetricpositivedefiniteB. ItcomputesthegeneralizedeigenvaluesofAandBusingtheCholeskyfactorizationofB.EIG(A,B,'qz')ignoresthesymmetryofAandBandusestheQZalgorithm.Ingeneral,thetwoalgorithmsreturnthesameresult,howeverusingtheQZalgorithmmaybemorestableforcertainproblems.TheflagisignoredwhenAandBarenotsymmetric.Seealsocondeig,eigs,ordeig.Overloadedmethods:ReferencepageinHelpbrowserdoceig -Classicsymmetriceigenvaluetest -Wilkinson'seigenvaluetest -Diagonalscalingtoimproveeigenvalue -Conditionnumberwithrespectto -Eigenvaluesand -Eigenvaluesofquasitriangular -ReordereigenvaluesinQZ -ReordereigenvaluesinSchur -Polynomialeigenvalue -Cmex-interfacetoARPACKlibraryusedby -Findafeweigenvaluesandeigenvectorsofamatrixusing -EIGShelper -Graphicaldemonstrationofeigenvaluesandsingular -Matrixexponentialviaeigenvaluesand -FindtheeigenvalueoftheMathieu's -Provideswidth&heightadjustmentsfor -neighbor-joiningmethodforphylogenetictree -classifiesdatausingthenearest-neighbor -imputesmissingdatausingthenearest-neighbor -calculatesmolecularweightfora -Biterrorrate(BER)forRayleighandRicianfadingchannels. -Calculatetheminimumdistanceofalinearblockcode. -ConstructaRayleighfadingchannel -Selectequalizerstructureandweightupdatealgorithm. -COMMBLKFADINGCHANRayleigh/Ricianchannelhelper -MaskdynamicdialogfunctionforRayleighnoise -COMMBLKRAYLEIGHCHANRayleighchannelhelper -Sortcomplexdiscreteeigenvaluesindescending -Sortcomplexcontinuouseigenvaluesindescending -Linearquadraticregulatordesignwithoutputweighting -Linear-quadraticregulatordesignwithoutput -Lyapunovequationsolutionusingeigenvalue -Assetcovariancefromreturnserieswithexponential -Convertportfolioholdingsintoportfolio -Randomizedportfoliorisks,returns,and -Portfolioweight -Convertportfoliovalueandweightsintoportfolioholdings. -calculatesthe(weighted)movingaverageofavectorof -Weighted -WeightedAverageLifeofmortgagepool. -Weightedaverageoftheparents. -NeighborhoodoperationsusinglookuptablesEigenvaluesandeigenvectorsd=d=eig(A,B)[V,D]=eig(A)d=eig(A)returnsavectoroftheeigenvaluesofmatrixd=eig(A,B)returnsavectorcontainingthegeneralizedeigenvalues,ifAandBaresquare IfSissparseandsymmetric,youcanused=eig(S)toreturntheeigenvaluesofS.IfSissparsebutnotsymmetric,orifyouwanttoreturntheeigenvectorsofS,usethefunctioneigsinsteadof[V,D]=eig(A)producesmatricesofeigenvalues(D)andeigenvectors(V)ofmatrixA,sothat=V*D.MatrixDisthecanonicalformofA—adiagonalmatrixwithA'seigenvaluesonthemaindiagonal.MatrixVisthemodalmatrix—itscolumnsaretheeigenvectorsofA.IfWisamatrixsuchthatW'*A=D*W',thecolumnsofWarethelefteigenvectorsofA.Use[W,D]=eig(A.');W=conj(W)tocomputethelefteigenvectors.[V,D]=eig(A,'nobalance')findseigenvaluesandeigenvectorswithoutapreliminarybalancingstep.Thismaygivemoreaccurateresultsforcertainproblemswithunusualscaling.Ordinarily,balancingimprovestheconditioningoftheinputmatrix,enablingmoreaccuratecomputationoftheeigenvectorsandeigenvalues.However,ifamatrixcontainssmallelementsthatarereallyduetoroundofferror,balancingmayscalethemuptomakethemassignificantastheotherelementsoftheoriginalmatrix,leadingtoincorrecteigenvectors.Usethenobalanceoptioninthisevent.Seethebalancefunctionformoredetails.[V,D]=eig(A,B)producesadiagonalmatrixDofgeneralizedeigenvaluesandafullmatrixVwhosecolumnsarethecorrespondingeigenvectorssothatA*V=B*V*D.[V,D]=eig(A,B,flag)specifiesthealgorithmusedtocomputeeigenvaluesandeigenvectors.flagcanbe:ComputesthegeneralizedeigenvaluesofAandBusingtheCholeskyfactorizationofB.Thisisthedefaultforsymmetric(Hermitian)Aandsymmetric(Hermitian)positivedefiniteB.Ignoresthesymmetry,ifany,andusestheQZalgorithmasitwouldfornonsymmetric(non-Hermitian)AandB. Foreig(A),theeigenvectorsarescaledsothatthenormofeachis1.0.Foreig(A,B),eig(A,'nobalance'),andeig(A,B,flag),theeigenvectorsarenotnormalized.AlsonotethatifAissymmetric,eig(A,'nobalance')ignoresthenobalanceoptionsinceAisalreadybalanced.Theeigenvalueproblemistodeterminethenontrivialsolutionsofthe isann-by-nmatrix, isalengthncolumnvector,and isascalar.Thenvaluesof satisfytheequationaretheeigenvalues,andthecorrespondingvaluesof aretherighteigenvectors.Thefunctioneigsolvesfortheeigenvalues,andoptionallytheeigenvectors.Thegeneralizedeigenvalueproblemistodeterminethenontrivialsolutionsofthewhereboth aren-by-nmatricesand isascalar.Thevaluesof thatsatisfytheequationarethegeneralizedeigenvaluesandthecorrespondingvaluesof arethegeneralizedright isnonsingular,theproblemcouldbesolvedbyreducingittoastandardeigenvalue canbesingular,analternativealgorithm,calledtheQZmethod,isWhenamatrixhasnorepeatedeigenvalues,theeigenvectorsarealwaysindependentandtheeigenvectormatrixVdiagonalizestheoriginalmatrixAifappliedasasimilaritytransformation.However,ifamatrixhasrepeatedeigenvalues,itisnotsimilartoadiagonalmatrixunlessithasafull(independent)setofeigenvectors.Iftheeigenvectorsarenotindependentthentheoriginalmatrixissaidtobedefective.Evenifamatrixisdefective,thesolutionfromeigsatisfiesA*X=TheB=[