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1、3NormalMatricesandSystems3.1NormalMatrices3.1.1DefinitionoftheNormalMatrixConsiderarationalmatrixinthestandardformL()sW()s=,(3.1.1)ms()mìnwhereL(s)≠[s]isapolynomialmatrixandm(s)(amonicpolynomial)istheleastcommondenominatoroftheentriesofthematrixW(s).Weassumethatthenumberofrowsmandcolumnsnofthematri
2、x(3.1.1)isgreaterthanorequaltotwo(thatis,m,ní2).Definition3.1.1.Arationalmatrixoftheform(3.1.1)iscallednormalifandonlyifeverynonzerosecond-orderminorofthepolynomialmatrixL(s)isdivisible(withoutremainder)bythepolynomialm(s).Forexample,thematrix»º10…»s+11»ºs+20L()sW()s==…»…»=(3.1.2)…»10()s++12()sm¬¼s
3、+1()s0…»¬¼s+2isnormal,sincethedeterminantofthematrix164PolynomialandRationalMatrices»ºs+20L()s=…»,(3.1.3)¬¼01s+whichisasecond-orderminorofthismatrix,isdivisible(withoutremainder)bythepolynomialm(s)=(s+1)(s+2).Ontheotherhand,thematrix»ºs+20…»2()s+11»ºs+20L()sW()s==…»…»=(3.1.4)2…»1()s+1¬¼01s+ms()…»0¬
4、¼s+1isnotnormal,sincethedeterminantofL(s)isnotdivisible(withoutremainder)by2thepolynomialm(s)=(s+1).-13.1.2NormalityoftheMatrix[Is–A]foraCyclicMatrix-1nìnTheinversematrix[Is–A]foranymatrixA≠isarationalmatrix,whichcanbewritteninthestandardform-1LA()s[]IAs-=,(3.1.5)ms()nìnwhereLA(s)≠[s]andm(s)isaleas
5、tcommondenominator.Applyingelementaryoperationsonrowsandcolumns,wecanreduce[Is–A]toitsSmithcanonicalform[IAUIAVss-=]()[s-](s)=S(3.1.6)nnì=≠diag»º¬¼isis12(),(),...,isr(),0,...,0<[]s,whereU(s)andV(s)areunimodularmatricesofelementaryoperationsonrowsandcolumns;i1(s),i2(s),…,ir(s)arethemonicinvariantpol
6、ynomialssatisfyingthedivisibilityconditionik+1(s)
7、ik(s)(thepolynomialik+1(s)isdivisiblewithoutremainderbythepolynomialik(s),k=1,…,r-1,andr=rankLA(s)).TheinvariantpolynomialsaregivenbytheformulaDsk()isk()===,fork1,...,,rDs()0()1,(3.1.7)Dsk-1()whereDk(s)isagreatestcommondivisorofallk-thorderminorsof[
8、Is–A].nìnTheminimalpolynomialY(s)ofamatrixA≠isrelatedtoitscharacteristicpolynomialj(s)=det[Ins–A]inthefollowingwayNormalMatricesandSystems165j()sY=()s.(3.1.8)Dsn-1()From(3.1.7)and(3.1.8)itfollowsthatY(s)=j(
