matrix polynomial equations, and rational and algebraic matrix equations

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1、6MatrixPolynomialEquations,andRationalandAlgebraicMatrixEquations6.1UnilateralPolynomialEquationswithTwoVariables6.1.1ComputationofParticularSolutionstoPolynomialEquationsConsiderthefollowingequationAXBY+=C,(6.1.1)lìplìqlìmpìmwhereA=A(s)≠[s],B=B(s)≠[s],C=C(s

2、)≠[s],X=X(s)≠[s]qìmandY=Y(s)≠[s].GiventhematricesA,BandC,computematricesXandYsatisfying(6.1.1).Thefollowingproblemwillbecalledthedualtotheaboveone.GiventhepolynomialmatricespmìììqmlmAA=≠()s---[],sssssBB=≠()[],CC=≠()[],lìplìqcomputepolynomialmatricesX=X(s)≠[s

3、]andY=Y(s)≠[s]satisfyingtheequationXA+=YBC.(6.1.2)Usingthetransposewecantransform(6.1.2)into(6.1.1).Theorem6.1.1.Equation(6.1.1)hasasolutionifandonlyifoneofthefollowingconditionsismet:1.[A,B,C]and[A,B,0]arerightequivalentmatrices,314PolynomialandRationalMatr

4、ices2.agreatestcommonleftdivisor(GCLD)ofthematricesAandBisaleftdivisorofthematrixC.Proof.LetX0,Y0beasolutionto(6.1.1),thatis,AX0+BY0=C.Then»IX00º…»[,,]ABC=+[]ABAXBY,,00=[,,0]0AB…IY0».…¬00I»¼AccordingtoDefinition1.7.1[A,B,C]and[A,B,0]arerightequivalentmatrice

5、s,since»ºIX00…»0IY(6.1.3)…»0…»¬¼00Iisaunimodularmatrix.Conversely,if[A,B,C]and[A,B,0]arerightequivalentmatrices,thenthereexistsaunimodularmatrixP=P(s)suchthat[,,][,,0]ABCABP=,(6.1.4)wherethematrixPisoftheform»ºIR01…»0IR.(6.1.5)…»2…»¬¼00IFrom(6.1.4)itfollowst

6、hatAR1+BR2=C.ThusthepairR1,R2constitutesasolutionto(6.1.1).Nowwewillshowthatif(6.1.1)hasthesolutionX0,Y0,thenGCLDofthematricesAandBisaleftdivisorofthematrixC.LetLbeaGCLDofthematricesAandB,thatis,AL==ABL,B,(6.1.6)11whereA1,B1arepolynomialmatrices.Substitution

7、of(6.1.6)intotheequationAX+=BYC(6.1.7)00yieldsMatrixPolynomialEquations,andRationalandAlgebraicMatrixEquations315LAXBY(10+=10)C.(6.1.8)ThusthematrixLisaleftdivisorofthematrixC.NowwewillshowthatifLisaleftdivisorofC,then(6.1.1)hasasolution.ByassumptionC=LC1,wh

8、ereC1isapolynomialmatrix.Ontheotherhand,theassumptionthatLisaGCLDofAandBimpliestheexistenceofpolynomialmatricesU11andU21suchthatAU+=BUL.(6.1.9)1121Post-multiplying(6.1.9)byC1,andtakingintoaccoun