(完整版)矩阵的特征值与特征向量-英文文献

6.2DefinitionsandexamplesDEFINITION6.1.1(Eigenvalue,eigenvector)LetAbeacomplexsquarematrix.ThenifisacomplexnumberandXanon—erocomplexcolumnvectorsatisfyingAXX,wecallXaneigenvectorofA,whileiscalledaneigenvalueofA.WealsosaythatXisaneigenvectorcorrespondingtotheeigenvalue?.SointheaboveexampleP1andP2areeigenvectorscorrespondingto1and2,respectively.WeshallgiveanalgorithmwhichstartsfromtheeigenvaluesofahtAandconstructsarotationmatrixAsuchthatPAPisdiagonal.hbAsnotedabove,ifisaneigenvalueofannnmatrixA,withcorrespondingeigenvectorX,then(AIn)X0,withX0,sodet(AIn)0andthereareatmostndistincteigenvaluesofA.Converselyifdet(AIn)0,then(AIn)X0hasanon-rivialsolutionXandso，isaneigenvalueofAwithXacorrespondingeigenvector.DEFINITION6.1.2(Characteristicpolynomial,equation)Thepolynomialdet(AIn)iscalledthecharacteristicpolynomialofAandisoftendenotedbychA().Theequationdet(AIn)0iscalledthecharacteristicequationofA.HencetheeigenvaluesofAaretherootsofthecharacteristicpolynomialofA.abFora22matrixA,itiseasilyverifiedthatthecharacteristicpolynomialiscdOurnextresulthaswideapplicability:THEOREM6.2.1LetAbea22matrixhavingdistincteigenvalues1and2andcorrespondingeigenvectorsX1andX2.LetPbethematrixwhosecolumnsareX1andX2,respectively.ThenPisnon-singularandP1APProof.SupposeAX11X1andAX22X2.WeshowthatthesystemofhomogeneousequationsxX1yX20hasonlythetrivialsolution.Thenbytheorem2.5.10thematrixP%.x2isnon-singular.SoassumexX1yX20.(6.3)ThenA(xX1yX2)A00,sox(AX1)y(g)0.Hencex1X1y2X20.(6.4)Multiplyingequation6.3byandsubtractingfromequation6.4gives1112bethematrixofexample6.2.1.Then1andX2areeigenvectorscorrespondingtoeigenvalues1immediateapplicationsoftheorem6.1.1.ThefirstistothecalculationofAn:If1PAPdiag(1,2),thenAThesecondapplicationistosolvingasystemoflineardifferentialequationsHencex11x1andy11y1.Thesedifferentialequationsarewell—nowntohavethesolutionsxixi(0)eltandyiyOeJwhere论(0)isthevalueof论whent0.kx,wherekisaconstant,thend/ktktktdyktkt.(ex)kexekexekx0.dtdtHenceektxisconstant,soektxek0x(0)x(0).Hencexx(0)ekt.]termsofx(0)andy(0).Henceultimatelyxandyaredeterminedasexplicitfunctionsoft,usingtheequationXPY.23EXAMPLE6.1.3LetA.Usetheeigenvaluemethodto45deriveanexplicitformulaforAnandalsosolvethesystemofdifferentialequationsdx2x3ydtdydt4x5y,givenx7andy13whent0.Solution.ThecharacteristicpolynomialofA.3is32whichhasdistinet131andX2.HenceifP,wehavePAPdiag(1,2)14An(Pdiag(1,2)P1)nPdiag((1)n,(2)n)P113(1)n043140(2)n11n131043Hence(1)n140213Tosolvethedifferentialequationsystem,makethesubstitutionXPYThenx捲3y「yx14y1.ThesystemthenbecomesX1xi?soy2y1为x1(0)etandY1%(o)e2t.Now.%(0)门1x(0)43711t2tP.S0X111eandY12e,Hence%（o）y(0)11136为11et3(6e2t)11et2t18e,y111et4(6e2t)11et24eForamorecomplicatedexamplewesolveasystemofinhomogeneousrecurreneerelations.EXAMPLE6.2.4SolvethesystemofrecurreneerelationsX2xnyn1yn1Xn2yn2'giventhatx00andy00.Solution.ThesystemcanbewritteninmatrixformasXn1AXnB,211WhereAandB.122ItisthenaneasyinductiontoprovethatXnAnX0(An1LAI2)B.(6.5)AlsoitiseasytoverifybytheeigenvaluemethodthatAn113n13n1u$v213n13n221111whereUandV.Hence1111j.(3n1LThenequation6.5giveswhichsimplifiesto1REMARK6.2.1If(A12)existed(thatis,ifdet(A12)0,orequivalently,if1isnotaneigenvalueofA),thenwecouldhaveusedtheformulan1n1An1LAI2(AnI2)(AI2)1.(6.6)HowevertheeigenvaluesofAare1and3intheaboveproblem,soformula6.6cannotbeusedthere.Ourdiscussionofeigenvaluesandeigenvectorshasbeenlimitedto22matrices.Thediscussionismorecomplicatedformatricesofsizegreaterthantwoandisbestlefttoasecondcourseinlinearalgebra.Neverthelessthefollowingresultisausefulgeneralizationoftheorem6.2.1.Thereaderisreferredto[28,page350]foraproof.THEOREM6.2.2LetAAbeannnmatrixhavingdistincteigenvalues1丄nandXn.LetPbethematrixwhosecolumnsarerespectivelyX1,LXn.ThenPisnon—ingularand100010200P1APMMMM00LnAnotherusefulresultwhichcoversthecasewheretherearemultipleeigen-valuesisthefollowing(Thereaderisreferredto[28,pages351—52]foraproof):THEOREM6.1.3SupposethecharacteristicpolynomialofAhasthefactorizationdet(In1)(G)n1L(ct)nt,whereg丄,ctarethedistineteigenvaluesofA.Supposethatfori1丄,t,wehavenullityginn)ni.Foreachsuchi,chooseabasisXi1,L,XinifortheeigenspaceN(ciInn).ThenthematrixPX11LX1niLXt1LXtnisnon-ingularandP-1APisthefollowingdiagonalmatrixGin0001P1AP0C2ln200MMMM00Lct1nt(Thenotationmeansthatonthediagonaltherearenrelementsg,followedbyn2elementsc2,...,ntelementsct.)