The Exponential of a Matrix

The first thing I need to do is to make sense of the matrix exponential $e^{At}$ ... You can compute the exponential of an arbitrary diagonal matrix in the ... TheExponentialofaMatrix Thesolutiontotheexponentialgrowthequation Itisnaturaltoaskwhetheryoucansolveaconstantcoefficient linearsystem inasimilarway. Ifasolutiontothesystemistohavethesameformasthegrowth equationsolution,itshouldlooklike ThefirstthingIneedtodoistomakesenseofthematrixexponential. TheTaylorseriesforis Itconvergesabsolutelyforallz. ItAisanmatrixwithrealentries,define Thepowersmakesense,sinceAisasquarematrix.It ispossibletoshowthatthisseriesconvergesforalltandevery matrixA. Differentiatingtheseriesterm-by-term, Thisshowsthatsolvesthedifferentialequation .Theinitialconditionvector yieldstheparticularsolution Thisworks,because(by settinginthepowerseries). Anotherfamiliarpropertyofordinaryexponentialsholdsforthe matrixexponential:IfAandBcommute(thatis,),then Youcanprovethisbymultiplyingthepowerseriesforthe exponentialsontheleft.(isjustwith.) Example.Computeif ComputethesuccessivepowersofA: Therefore, Youcancomputetheexponentialofanarbitrarydiagonalmatrixin thesameway: Example.Computeif ComputethesuccessivepowersofA: Hence, Here'swherethelastequalitycamefrom: Example.Compute,if IfyoucomputepowersofAasinthelasttwoexamples,thereisno evidentpattern.Therefore,itwouldbedifficulttocomputethe exponentialusingthepowerseries. Instead,setupthesystemwhosecoefficientmatrixisA: Thesolutionis Next,notethatifBisamatrix, Inparticular,thisistruefor.Now isthesolutionsatisfying,but Settogetthefirstcolumnof: Hence,,.So Settogetthesecondcolumnof: Therefore,,.Hence, Therefore, Ifound,butIhadtosolveasystemof differentialequationsinordertodoit. Insomecases,it'spossibletouselinearalgebratocomputethe exponentialofamatrix.AnmatrixAisdiagonalizableifithasnindependent eigenvectors.(Thisistrue,forexample,ifAhasndistinct eigenvalues.) SupposeAisdiagonalizablewithindependenteigenvectorsandcorrespondingeigenvalues.LetSbethematrixwhose columnsaretheeigenvectors: Then AsIobservedabove, Ontheotherhand,since, Hence, IcanusethisapproachtocomputeincaseAisdiagonalizable. Example.Computeif Theeigenvaluesare,.Sincetherearetwodifferenteigenvalues andAisamatrix,Aisdiagonalizable.The correspondingeigenvectorsareand.Thus, Hence, Example.Computeif Theeigenvaluesareand(double).Thecorrespondingeigenvectorsarefor,andandfor.SinceIhave3independenteigenvectors,thematrix isdiagonalizable. Ihave Fromthis,itfollowsthat Here'saquickcheckonthecomputation:Ifyousetintherightside,youget Thischecks,since. Notethatthischeckisn'tfoolproof---justbecauseyougetIby settingdoesn'tmeanyouranswerisright.However, ifyoudon'tgetI,youranswerissurelywrong! HowdoyoucomputeisAisnotdiagonalizable? I'lldescribeaniterativealgorithmforcomputingthatonlyrequiresthatoneknowtheeigenvaluesof A.Therearevariousalgorithmsforcomputingthematrixexponential; thisone,whichisduetoWilliamson[1],seemstometobethe easiestforhandcomputation. (Notethatfindingtheeigenvaluesofamatrixis,ingeneral,a difficultproblem:Anymethodforfindingwillhavetodealwithit.) LetAbeanmatrix.Letbealistofthe eigenvalues,withmultipleeigenvaluesrepeatedaccordingtotheir multiplicity. Let Then Toprovethis,I'llshowthattheexpressionontherightsatisfies thedifferentialequation.To dothis,I'llneedtwofactsaboutthecharacteristicpolynomial. 1.. 2.(Cayley-HamiltonTheorem). Observethatifisthecharacteristicpolynomial, thenusingthefirstfactandthedefinitionoftheB's, BytheCayley-HamiltonTheorem, Iwillusethisfactintheproofbelow. Example.I'llillustratetheCayley-Hamilton theoremwiththematrix Thecharacteristicpolynomialis.TheCayley-Hamiltontheorem assertsthatifyouplugAinto, you'llgetthezeromatrix. First, Therefore, Proofofthealgorithm.First, RecallthattheFundamentalTheoremofCalculussaysthat ApplyingthisandtheProductRule,Icandifferentiatetoobtain Therefore, Expandthetermsusing Makingthissubstitutionandtelescopingthesum,Ihave (Theresult(*)provedabovewasusedinthenext-to-the-last equality.)Combiningtheresultsabove,I'veshownthat Thisshowsthatsatisfies. Usingthepowerseriesexpansion,Ihave.So (Rememberthatmatrixmultiplicationisnotcommutativeingeneral!) Itfollowsthatisaconstantmatrix. Set.Since,itfollowsthat.Inaddition,. Therefore,,andhence. Example.Usethematrixexponentialtosolve Thecharacteristicpolynomialis.Youcan checkthatthereisonlyoneindependenteigenvector,soIcan't solvethesystembydiagonalizing.Icoulduse generalizedeigenvectorstosolvethesystem,butIwillusethe matrixexponentialtoillustratethealgorithm. First,listtheeigenvalues:.Sinceisadoubleroot,itislistedtwice. First,I'llcomputethe's: Herearethe's: Therefore, Asacheck,notethatsettingproducesthe identity.) Thesolutiontothegiveninitialvalueproblemis Youcangetthegeneralsolutionbyreplacingwith. Example.Findif Theeigenvaluesareobviously(double)and . First,I'llcomputethe's.Ihave,and Next,I'llcomputethe's.,and Therefore, Example.Usethematrixexponentialtosolve Thisexamplewilldemonstratehowthealgorithmforworkswhentheeigenvaluesarecomplex. Thecharacteristicpolynomialis. Theeigenvaluesare.I willlistthemas. First,I'llcomputethe's.,and Next,I'llcomputethe's.,and Therefore, Iwantarealsolution,soI'lluseDeMoivre'sFormulato simplify: Pluggingtheseintotheexpressionforabove,Ihave Noticethatallthei'shavedroppedout!Thisreflectstheobvious factthattheexponentialofarealmatrixmustbearealmatrix. Finally,thegeneralsolutiontotheoriginalsystemis Example.I'llcomparethematrixexponential andtheeigenvectorsolutionmethodsbysolvingthefollowingsystem bothways: Thecharacteristicpolynomialis. Theeigenvaluesare. Consider: Asthisisaneigenvectormatrix,itmustbesingular,andhencethe rowsmustbemultiples.Soignorethesecondrow.Iwantavector suchthat.Togetsuchavector,switchtheand-1andnegateoneofthem:,.Thus,isaneigenvector. Thecorrespondingsolutionis Taketherealandimaginaryparts: Thesolutionis NowI'llsolvetheequationusingtheexponential.Theeigenvalues are.Computethe's.,and (Hereandbelow,I'mcheatingalittleinthecomparisonbynot showingallthealgebrainvolvedinthesimplification.Youneedto useDeMoivre'sFormulatoeliminatethecomplexexponentials.) Next,computethe's.,and Therefore, Thesolutionis TakingintoaccountsomeofthealgebraIdidn'tshowforthematrix exponential,Ithinktheeigenvectorapproachiseasier. Example.Solvethesystem Forcomparison,I'lldothisfirstusingthegeneralizedeigenvector method,thenusingthematrixexponential. Thecharacteristicpolynomialis. Theeigenvalueis(double). Ignorethefirstrow,anddividethesecondrowby2,obtainingthe vector.Iwantsuchthat.Swap1 and-2andnegatethe-2:Iget.Thisis aneigenvectorfor. SinceIonlyhaveoneeigenvector,Ineedageneralizedeigenvector. ThismeansIneedsuchthat Rowreduce: Thismeansthat.Settingyields.Thegeneralized eigenvectoris. Thesolutionis Next,I'llsolvethesystemusingthematrixexponential.The eigenvaluesare.First,I'llcomputethe's.,and Next,computethe's.,and Therefore, Thesolutionis Inthiscase,findingthesolutionusingthematrixexponentialmay bealittlebiteasier. [1]RichardWilliamson,Introductiontodifferential equations.EnglewoodCliffs,NJ:Prentice-Hall,1986. Sendcommentsaboutthispageto: [email protected]. BruceIkenaga'sHomePage Copyright2012byBruceIkenaga