Hankel矩阵的性质及其应用

204()Vol.20No.4200511JOURNALOFZHENGZHOUUNIVERSITYOFLIGHTINDUSTRY(NaturalScience)Nov.2005:2005-07-11:(1966),,,,,:.:1004-1478(2005)04-0097-03Hankel1,2(11,450002;21,450004):Bez(a,b)a(),a()b()Hankel;BezoutBarnettBez(a,b)a(),Hankel.:Hankel;;;Barnett;:O151121:APropertiesandapplicationofHankelmatrixTANRui-mei1,SHICheng-tang2(11Dept.ofInfor.andCalc.Sci.,ZhengzhouUniv.ofLightInd.,Zhengzhou450002,China;21Dept.ofEconomicTrade,ZhengzhouElectr.PowerCollege,Zhengzhou450004,China)Abstract:SomenewcongruenceandtwiningrelationsareinvestigatedbyusingtheknownoneofBez(a,b)withthefirstcompanionmatrixofa().Anewmethodtojudgetherelativelyprimeondoublepolynomialsa()andb()isgivenintermsoftheBarnettfactoritionofBezoutmatrixandtheunanimityofthezero-pointswiththeeigenvaluesofthefirstcompanionmatrixofa().Keywords:Hankelmatrix;symmetrizer;companionmatrix;Barnettfactorization;relativelyprimeofpolynomials0HankelBezout1,Hankel;dx(t)dt=Ax(t)+u(t)b,y(t)=CTx(t)Hankel.Bez(a,b)a(),a()b()Hankel;BezoutBar2nettBez(a,b)a(),Hankel.11[1]a()=nj=0ajj(an0)b()=nj=0bjj.||,b()/a()b()/a()=j=0hj-j,h0,h1,h2,b()/a()Markov.{hj}j=0Hankel[hi+j-1]i,j=1.,Hn(b/a)=[hi+j-1]ni,j=1.2[2]a()=nj=0ajj(an0),b()=nj=0bjj,[a()b()-a()b()]/(-)=n-1i,j=0dijijB(a,b)=(dij)n-1i,j=0Mn(C)a(),b()Bezout.3[2]a()=nj=0ajj,HankelS(a)=a1a2a3an-1ana2a3a4an0a3a4a500an0000Mn(C)a().an0,detS(a)0,nCa=010000100001-a0an-a1an-a2an-an-1anMn(C)a()=nj=0ajj(an0)().HankelBezout.1[1]a()=nj=0ajj(an0)b()=nj=0bjj,B(a,b)=S(a)HS(a),H=[hi+j-1]ni,j=1Mn(C)b()/a()MarkovHankel,S(a)a().2[2]a()b()1,BezoutB(a,b)BarnettB(a,b)=S(a)b(Ca)H=Hn(b/a)=b(Ca)S-1(a)2Hankel3a()b()1,a=a()=a(+),b=b()=b(+),Hba=TtHbaT,C,T=S(a)(V(n))tS(a)-1,V(n)=jij-in-1i,j=0Mn(C),ji,ji=0.Bez(a,b)=V(n)Bez(a,b)(V(n))tS(a)HbaS(a)=Bez(a,b)=V(n)Bez(a,b)(V(n))t=V(n)S(a)HbaS(a)(V(n))tHba=S(a)-1V(n)S(a)HbaS(a)(V(n))tS(a)-1T=S(a)(V(n))tS(a)-1,Hba=TtHbaT4a()=nj=0ajj,…a()=nj=0…ajj,H…aa3=Ha…aHa…a3=H…aa(B(a,…a))3=-B(a,…a)=B(…a,a)=S(…a)Ha…aS(…a)H…aa=S(a)-1B(a,…a)S(a)-1H…aa3=[S(a)-1]3B(a,…a)3[S(a)-1]3=[S(a)3]-1(-B(a,…a))[S(a)3]-1=S(…a)-1B(…a,a)S(…a)-1=Ha…a5a(),b(),Hba1,H^b^a=-TtHbaT89()2005,T=S(a)PS(^a)-1n;Pn;^a=^a()=na(-1),^b=^b()=nb(-1).B(^a,^b)=-PB(a,b)PS(^a)H^b^aS(^a)=-PS(a)HbaS(a)PH^b^a=-S(^a)-1PS(a)HbaS(a)PS(^a)-1=-[S(a)PS(^a)-1]tHba[S(a)PS(^a)-1]T=S(a)PS(^a)-1,.3HankelHankel,Hankel.6a()=nj=0ajj(an0),b()=nj=0bjj,detH=detHn(b/a)0,a()b().:an0,detS(a)0,detH=detHn(b/a)=det(b(Ca)S-1(a))=detb(Ca)detS-1(a)0,detb(Ca)0.a()1,2,,n,CaMn(C).b(Ca)b(1),b(2),,b(n),detb(Ca)=b(1)b(2)b(n)01,2,,nb(),a()b().,.3,Hankel.7a()=nj=0ajj(an0),b()=nj=0bjj,c()=nj=0cjj,,C,H((b()+c())/a())=H(b/a)+H(c/a)H((b+c)/a)=(b+c)(Ca)S-1(a)=(b(Ca)+c(Ca))S-1(a)=b(Ca)S-1(a)+c(Ca)S-1(a)=H(b/a)+H(c/a).8a()b()6,(a(),b())=1,(a()+b(),a())=1.6,detH((a+b)/a)=det[H(b/a)+H(a/a)]=detH(b/a)0,.:[1].[M].:,1990.187189.[2]ChenGongning,YangZhenghong.BezoutianrepresentationviaVandermondematrices[J].LinearAlgebraAppl,1993,186:37441[3].[M].3.:,2004.20053,,32005,DNA,,.()994:Hankel