矩阵的特征值与特征向量-英文文献(可编辑修改word版)

1、6.2DefinitionsandexamplesDEFINITION6.LI(Eigenvaluc.cigenvector)LetAbeacomplexsquarematrix.ThenifisacomplexnumberandXanon-zerocomplexcolumnvectorsatisfyingAX=X,wecallXaneigenvectorA,whileiscalledaneigenvalueofA.WcalsosaythatofXisaneigenvectorcorrespondingtotheeigenvalueSointheaboveexampleHandarceigen

2、vectorscorrespondingto)and2,respectively.WcshallgiveanalgorithmwhichstartsfromtheeigenvaluesofA=hbandconstructsarotationmatrixAsuchthatPAPisdiagonal.Asnotedabove,ifisaneigenvalueofannxnmatrixA,withcorrespondingeigenvectorX9then(A-In)X=0,withXH0,sodet(A一/“)=0andthereareatmostndistincteigenvaluesofA.C

3、onverselyifdet(A一/)=0.then(A-/n)X=0hasanon-trivialsolutionXand50, isaneigenvalueofAwithXacorrespondingeigenvector.DEFINITION6.L2(CharacteristicpolynomiaLequation)Thepolynomialdet(A-In)iscalledthecharacteristicpolynomialofAandisoftendenotedbychA().Theequationdet(A一/)=0iscalledthecharacteristicequatio

4、nofA.HencetheeigenvaluesofAarctherootsofthecharacteristicpolynomialofA.a加Fora2x2matrixA=,itiseasilyverifiedthatthecharacteristicpolynomialis(traceA)+detA,wheretraceA=a+disthesumofthediagonalelementsofA.2nEXAMPLE6.2.1FindtheeigenvaluesofA=Iandfindalleigen-vectors.12)Solution.Thecharacteristicequation

5、ofAis2-4+3=0,or(D(-3)=0.Hence=1or3.Theeigenvectorequation(A-In)X=0reducestoor2-1nr2-(2-)x+y=Ox+(2-)y=0Taking=1givesx+y=Ox+y=Owhichhassolutionx=-y,yarbitrary.Consequentlytheeigenvectorscorrespondingto),、=1arethevectorsII,with.0.9Taking=3gives-x+y=0Jxy=0.whichhassolutionx=-y,yarbitrary.Consequentlythe

6、eigenvectorscorre-自spendingto=3arcthevectorsII,withyW0.J凶Ournextresulthaswideapplicability:THEOREM6.2.1LetAbea2x2matrixhavingdistincteigenvalues】and2andcorrespondingeigenvectorsandX2.LetPbethematrixwhosecolumnsareandX2,respectively.ThenPisnon-singularand012Proof.SupposeAX1=aandAX2=22-Wcshowthatthesy

7、stemofhomogeneousequationsxX+yX2=0hasonlythetrivialsolution.Thenbytheorem2.5.10thematrixP=x/4isnonsingular.SoassumexX+yX2=0.(6.3)ThenA(xX+yX2)=AO=0,sox(AX)+y(AX2)=0.HencexXx+=0.(6.4)Multiplyingequation6.3by1andsubtractingfromequation6.4gives"WO.Hencey=O,as(2-J0andX2W0,Thenfromequation6.3,xX,=0a

8、ndhencex=0.ThentheequationsAX=andAX2=2X2giveAP=Ax/xi=Avi/Ac=xi/X2十.1_尸_1.17.1.J2nEXAMPLE6.2.2LetA=bethematrixofexample6.2.1.Thenarceigenvectorscorrespondingtoeigenvaluesrioi1and3,respectively.HenceifP=wehavePAP=Therearetwoimmediateapplicationsoftheorem6.1.1.ThefirstistothecalculationofAn:IfLAP=diag(

9、,).thenA=Pdiag(,)Pand121,Thesecondapplicationistosolvingasystemoflineardifferentialequationsdx,dy,=ax+bx=xc+dxdtdtJb-whereA=isamatrixofrealorcomplexnumbersandxandycdarefunctionsoft.ThesystemcanbewritteninmatrixformasX=AX,dxIY卜Ie=/Iyl_lLJUiWemakethesubstitutionX=PY,where丫=ThenxandyarcalsofunctionsE11

10、ofrand=,=AX=(P-AP)Y=Fi0y.XPY02An=(Pdiag(-l,-2)piy=P"吆(一1),(一2)kr：张力'L14JLJLHence=(-1J1=1311oir4-314_l|_033x24-3114x2"_|_-l1_4-3x2-3+3x244x23+4x2Tosolvethedifferentialequationsystem,makethesubstitutionX=PY.Thenx=X+3y,y=x,+4.Thesystemthenbecomes''一"A|sox=x(0)6一andy=y(0)2f.No

11、w.ii兑=-2yP七加2刊;3U%"Iso?=75and凹=2户,HenceLLLx=71/+3(6产)=一11,+18产,y=-11/+4(6«-)=一15+24e-2/Foramorecomplicatedexamplewesolveasystemofinhomogeneousrecurrencerelations.EXAMPLE6.2.4Solvethesystemofrecurrencerelations%=2x-另用=一与+2%+2giventhatx0=0and=0.Solution.ThesystemcanbewritteninmatrixformasA=W

12、here2-1L-i2JandXm=4X+8,-11-2匕Itisthenaneasyinductiontoprovethat(6.5)X/A“X0+(41+A+QB.Alsoitiseasytoverifybytheeigenvaluemethodthatll+31-3113A=_=_+5|J31+322nn17IwhereU=IandV=.Hence11-11.n(3"-141-3+1)A+人=-U+V.-22Thenequation6.5gives1301(3-1)F-11七二(产/7+(产+,mJIm_whichsimplifiestoxn'2+1-3,->&g

13、t;|_%12一5-3%_|Hencex=(In+1-3")/and);=(2-5+3"y4.REMARK6.2.1If(A-Af1existed(thatis,ifdet(A-A)W0,orequivalently,if1isnotaneigenvalueofA),thenwecouldhaveusedtheformula+A+4=(一A)(A一,2尸.(6.6)HowevertheeigenvaluesofAarc1and3intheaboveproblem,soformula6.6cannotbeusedthere.Ourdiscussionofeigenvalues

14、andeigenvectorshasbeenlimitedto2x2matrices.Thediscussionismorecomplicatedformatricesofsizegreaterthantwoandisbestlefttoasecondcourseinlinearalgebra.Neverthelessthefollowingresultisausefulgeneralizationoftheorem6.2.1.Thereaderisreferredto28,page350foraproof.THEOREM6.2.2LetAAbeannxnmatrixhavingdistinc

15、teigenvalues,andcorrespondingeigenvectorsX,LetPbethematrixwhosecolumnsarerespectivelyXi,ThenPisnoih-singularandfi0001P-lAP=02°° |o0JAnotherusefulresultwhichcoversthecasewheretherearemultipleeigenvaluesisthefollowing(Thereaderisreferredto28,pages351-352foraproof):THEOREM6.1.3SupposethecharacteristicpolynomialofAhasthefactorizationwherec>,qarethedistincteigenvaluesofA.Supposethatfori=1,wchavenullityq/“一