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1、CHAPTER18HowtoCalculatetheEigenvaluesofSelf-AdjointMatrices1.Thebasisofoneofthemosteffectivemethodsforcalculatingapproximatelytheeigenvaluesofaself-adjointmatrixisbasedontheQRdecomposition.Theorem1.EveryrealinvertiblesquarematrixAcanbefactoredasA=QR,(1)w
2、hereQisanorthogonalmatrixandRisanuppertriangularmatrixwhosediagonalentriesarepositive.Proof.ThecolumnsofQareconstructedoutofthecolumnsofAbyGram-Schmidtorthonormalization.SothejthcolumnqjofQisalinearcombinationofthefirstjcolumnsa1,...,ajofA:qt=cllal,q2=cl
3、2at+c22a2,etc.Wecaninverttherelationbetweentheq-sandthea-s:al=rllgt,a2=rt2gt+r22q2,(2)an=ringi++rnngnLinearAlgebraandItsApplications,SecondEdition,byPeterD.LaxCopyrightQ2007JohnWiley&Sons,Inc.262HOWTOCALCULATETHEEIGENVALUESOFSELF-ADJOINTMATRICES263SinceA
4、isinvertible,itscolumnsarelinearlyindependent.Itfollowsthatallcoefficientsri1,...,r,in(2)arenonzero.Wemaymultiplyanyofthevectorsqjby-1,withoutaffectingtheirorthonormality.Inthiswaywecanmakeallthecoefficientsri1,...,r,,,,in(2)positive.HereAisannxnmatrix,D
5、enotethematrixwhosecolumnsareqi,...,qbyQ,anddenotebyRthematrixforij.Relation(2)canbewrittenasamatrixproductA=QR.SincethecolumnsofQareorthonormal,Qisanorthogonalmatrix.Itfollowsfromthedefinition(3)ofRthatRisuppertriangular.SoA=QRisthesought-aft
6、erfactorization(1).Thefactorization(1)canbeusedtosolvethesystemofequationsAx=u.ReplaceAbyitsfactoredform,QRx=uandmultiplybyQTontheleft.SinceQisanorthogonalmatrix,QTQ=1,andwegetRx=QTu.(4)SinceRisuppertriangularanditsdiagonalentriesarenonzero,thesystemofeq
7、uationscanbesolvedrecursively,startingwiththenthequationtodeterminex,,,thenthe(n-1)stequationtodetermineandsoallthewaydowntoxi.InthischapterweshallshowhowtousetheQRfactorizationofarealsymmetricmatrixAtofinditseigenvalue.TheQRalgorithmwasinventedbyJ.G.F.F
8、rancisin1961;itgoesasfollows:LetAbearealsymmetricmatrix;wemayassumethatAisinvertible,forwemayaddaconstantmultipleoftheidentitytoA.FindtheQRfactorizationofA:A=QR.DefineAIbyswitchingthefactorsQandRAi=RQ.(5)264LINEARALGEBRAAN
