linearalgebraanditsapplications spectral theory

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1、CHAPTER6SpectralTheorySpectraltheoryanalyzeslinearmappingsofaspaceintoitselfbydecomposingthemintotheirbasicconstituents.Westartbyposingaproblemoriginatinginthestabilityofperiodicmotionsandshowhowtosolveitusingspectraltheory.Weassumethatthestateofthesystemunderstudycanbedescribedbyafini

2、tenumbernofparameters;thesewelumpintoasinglevectorxin1°.Second,weassumethatthelawsgoverningtheevolutionintimeofthesystemunderstudydetermineuniquelythestateofthesystematanyfuturetimeiftheinitialstateofthesystemisgiven.Denotebyxthestateofthesystemattimet=0;itsstateatt=1isthencompletelyde

3、terminedbyx;wedenoteitasF(x).WeassumeFtobeadifferentiablefunction.Weassumethatthelawsgoverningtheevolutionofthesystemarethesameatalltimes;itfollowsthenthatifthestateofthesystemattimet=1isz,itsstateattimet=2isF(z).Moregenerally,Frelatesthestateofthesystemattimettoitsstateatt+1.Assumetha

4、tthemotionstartingatx=0isperiodicwithperiodone,thatisthatitreturnsto0attimet=1.ThatmeansthatF(0)=0,(1)Thisperiodicmotioniscalledstableif,startingatanypointItsufficientlyclosetozero,themotiontendstozeroasttendstoinfinity.ThefunctionFdescribingthemotionisdifferentiable;thereforeforsmallh

5、,F(h)isaccuratelydescribedbyalinearapproximation:F(h)Ah.(2)ForpurposesofthisdiscussionweassumethatFisalinearfunctionF(h)=Ah,(3)LinearAlgebraandItsApplications.SecondEdition,byPeterD.LaxCopyright?"2007JohnWiley&Sons,Inc.58SPECTRALTHEORY59Aanitxnmatrix.Thesystemstartingathwill,aftertheel

6、apseofNunitsoftime,beinthepositionANh.(4)Inthenextfewpagesweinvestigatesuchsequences,thatis,oftheformh,Ah,...,ANh.....(5)FirstafewexamplesofhowpowersANofmatricesbehave;wechooseN=1024,becausethenANcanbeevaluatedperformingtensquaringoperations:Case(a)(b)3A(14)(5364)A'°24>10700

7、numericalexperimentsstronglysuggestthat(a)AN-ocasN-poo,(b)AN->0asNoo,thatis,eachentryofANtendstozero.Weturnnowtoatheoreticalanalysisofthebehaviorofsequencesoftheform(5).SupposethatavectorIt00hasthespecialpropertywithrespecttothematrixAthatAhismerelyamultipleofh:Ah=ah,whereaisascalara