functions of matrices and analytic similarity

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1、September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch03page123Chapter3FunctionsofMatricesandAnalyticSimilarity3.1ComponentsofaMatrixandFunctionsofMatricesInthischapter,weassumethatallthematricesarecomplexvalued(F=C)unlessotherwisestated.Let

2、φ(x)beapolynomial(φ∈C[x]).Thefollowingrelationsareeasilyestablishedφ(B)=Pφ(A)P−1,B=PAP−1,A,B∈Cn×n,P∈GL(C),n(3.1.1)φ(A1⊕A2)=φ(A1)⊕φ(A2).Itoftenpaystoknowtheexplicitformulaforφ(A)intermsoftheJordancanonicalformofA.Inviewof(3.1.1)itisenoughtoconsiderthecasewhereJis

3、composedofoneJordanblock.Lemma3.1.1LetJ=λ0I+H∈Cn×n,whereH=Hn.Thenforanyφ∈C[x]n
−1φ(k)(λ)0kφ(J)=H.k!k=0123September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch03page124124MatricesProof.ForanyφwehavetheTaylorexpansion
Nφ(k)(λ)0kφ(x)=(x−λ0),N=

4、max(degφ,n).k!k=0AsH
=0for≥nfromtheaboveequalitywededucethelemma.
UsingtheJordancanonicalformofAweobtain.Theorem3.1.2LetA∈Cn×n.AssumethattheJordancanon-icalformofAisgivenby(2.6.5).Thenforφ∈C[x]wehavem
ij−1(k)⊕
qiφ(λi)kP−1φ(A)=Pi=1⊕j=1Hmij.(3.1.2)k!k=0Definit

5、ion3.1.3LettheassumptionsofTheorem3.1.2hold.ThenZik=Zik(A)iscalledthe(i,k)componentofAandisgivenbyZ=P(0⊕···⊕0⊕qiHk⊕0···⊕0)P−1,ikj=1mij(3.1.3)k=0,...,si−1,si=mi1,i=1,...,.Compare(3.1.2)with(3.1.3)todeduce

s
i−1φ(j)(λ)iφ(A)=Zij.(3.1.4)j!i=1j=0Definition3.1.4LetA∈

6、Cn×nandassumethatΩ⊂Ccon-tainsspec(A).Thenforφ∈H(Ω)defineφ(A)by(3.1.4).Using(3.1.3)itiseasyverifythatthecomponentsofAsatisfyZij,i=1,...,,j=1,...,si−1,arelinearlyindependent,ZijZpq=0,ifeitheri=p,ori=pandj+q≥si,ZijZiq=Zi(j+q),forj+q≤si−1,

A=PλZ+ZP−1.ii0i1i=1(3.

7、1.5)September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch03page125FunctionsofMatricesandAnalyticSimilarity125ConsiderthecomponentZi(si−1).TheaboverelationsimplyAZi(si−1)=Zi(si−1)A=λiZi(si−1).(3.1.6)ThusthenonzerocolumnsofZi(s−1),Z
aretheeig

8、envectorsii(si−1)ofA,A
respectivelycorrespondingtoλi.(NotethatZi(s−1)=0.)iLemma3.1.5LetA∈Cn×n.AssumethatλiisaneigenvalueofA.LetXibethegeneralizedeigenspaceofAcorresp