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1、OnTaylor’sformulafortheresolventofacomplexmatrixMatthewX.Hea,PaoloE.Riccib,_Articlehistory:Received25June2007Receivedinrevisedform14March2008Accepted25March2008Keywords:PowersofamatrixMatrixinvariantsResolvent1.IntroductionAsaconsequenceoftheHilbertidentityin[1],theresolvent=ofanonsingularsquaremat
2、rix(denotingtheidentitymatrix)isshowntobeananalyticfunctionoftheparameterinanydomainDwithemptyintersectionwiththespectrumof.Therefore,byusingTaylorexpansioninaneighborhoodofanyfixed,wecanfindin[1]arepresentationformulaforusingallpowersof.Inthisarticle,byusingsomeprecedingresultsrecalled,e.g.,in[2],
3、wewritedownarepresentationformulausingonlyafinitenumberofpowersof.Thisseemstobenaturalsinceonlythefirstpowersofarelinearlyindependent.Themaintoolinthisframeworkisgivenbythemultivariablepolynomials(;)(see[2–6]),dependingontheinvariantsof);heremdenotesthedegreeoftheminimalpolynomial.2.Powersofmatrice
4、sandfunctionsWerecallinthissectionsomeresultsonrepresentationformulasforpowersofmatrices(seee.g.[2–6]andthereferencestherein).Forsimplicitywerefertothecasewhenthematrixisnonderogatorysothat.Proposition2.1.Letbeancomplexmatrix,anddenotebytheinvariantsof,andby.5itscharacteristicpolynomial(byconventio
5、n);thenforthepowersofwithnonnegativeintegralexponentsthefollowingrepresentationformulaholdstrue:.(2.1)Thefunctionsthatappearascoefficientsin(2.1)aredefinedbytherecurrencerelation,(2.2)andinitialconditions:.(2.3)Furthermore,ifisnonsingular,thenformula(2.1)stillholdsfornegativevaluesofn,providedthatw
6、edefinethefunctionfornegativevaluesofnasfollows:，.3.TaylorexpansionoftheresolventWeconsidertheresolventmatrixdefinedasfollows:.(3.1)NotethatsometimesthereisachangeofsigninEq.(3.1),butthisofcourseisnotessential.Itiswellknownthattheresolventisananalytic(rational)functionofineverydomainDofthecomplexpl
7、aneexcludingthespectrumof,andfurthermoreitisvanishingatinfinitysotheonlysingularpoints(poles)ofaretheeigenvaluesof.In[6]itisprovedthattheinvariantsofarelinkedwiththoseofbytheequations,.（3.2）Asaconsequenceof
