canonical forms for similarity

《canonical forms for similarity》由会员上传分享，免费在线阅读，更多相关内容在工程资料-天天文库。

1、September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch02page75Chapter2CanonicalFormsforSimilarity2.1StrictEquivalenceofPencilsDefinition2.1.1AmatrixA(x)∈D[x]m×nisapencilifA(x)=A+xA,A,A∈Dm×n.(2.1.1)0101ApencilA(x)isregularifm=nanddetA(x)=0.OtherwiseA(x)isas

2、ingularpencil.TwopencilsA(x),B(x)∈D[x]m×narestrictlyequivalentifsA(x)∼B(x)⇐⇒B(x)=QA(x)P,P∈GL(n,D),Q∈GL(m,D).(2.1.2)TheclassicalworksofWeierstrass[Wei67]andKronecker[Kro90]classifytheequivalenceclassesofpencilsunderthestrictequivalencerelationinthecaseDisafieldF.Wegiveashortacco

3、untoftheirmainresults.FirstnotethatthestrictequivalenceofA(x),B(x)impliestheequivalenceofA(x),B(x)overthedomainD[x].FurthermoreletB(x)=B0+xB1.(2.1.3)75September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch02page7676MatricesThenthecondition(2.1.2)isequivale

4、nttoB0=QA0P,B1=QA1P,P∈GL(n,D),Q∈GL(m,D).(2.1.4)ThuswecaninterchangeA0withA1andB0withB1withoutaffect-ingthestrictequivalencerelation.Hence,itisnaturaltoconsiderahomogeneouspencilA(x0,x1)=x0A0+x1A1.(2.1.5)AssumethatDisDU.ThenDU[x0,x1]isalsoDU(Problem1.4.6)InparticularDU[x0,x1]isD

5、G.Letδk(x0,x1),ik(x0,x1)betheinvariantdeterminantsandfactorsofA(x0,x1)respectivelyfork=1,...,rankA(x0,x1).Lemma2.1.2LetA(x0,x1)beahomogeneouspenciloverDU[x0,x1].Thenitsinvariantdeterminantsandtheinvariantpoly-nomialsδk(x0,x1),ik(x0,x1),k=1,...,rankA(x0,x1)arehomo-geneouspolyno

6、mials.Moreover,ifδk(x)andik(x)aretheinvariantdeterminantsandfactorsofthepencilA(x)fork=1,...,rankA(x),thenδk(x)=δk(1,x),ik(x)=ik(1,x),k=1,...,rankA(x).(2.1.6)Proof.Clearlyanyk×kminorofA(x0,x1)iseitherzeroorahomogeneouspolynomialofdegreek.InviewofProblem1wededucethattheg.c.d.of

7、allnonvanishingk×kminorsisahomogeneouspolynomialδ(x,x).Asi(x,x)=δk(x0,x1)Problem1impliesk01k01δk−1(x0,x1)thatik(x0,x1)isahomogeneouspolynomial.ConsiderthepencilA(x)=A(1,x).Soδk(x)—theg.c.d.ofk×kminorsofA(x)isobvi-ouslydivisiblebyδk(1,x).Ontheotherhand,wehavethefollowingrelatio

8、nbetweentheminorsofA(x0,x1)andA(x)
kx1detA(x0,x1)[α,β]=x0det