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1、2RationalFunctionsandMatrices2.1BasicDefinitionsandOperationsonRationalFunctionsAquotientoftwopolynomialsl(s)andm(s)invariables,wherem(s)isanonzeropolynomial,ls()ws()=(2.1.1)ms()iscalledarationalfunctionofthevariables.Thesetofrationalfunctionswithcoefficientsfromafieldwillbedenotedby(s).Afieldca
2、nbethefieldofrealnumbers,ofthecomplexnumbers,oftherationalnumbers,orafieldofrationalfunctionsofanothervariablez,etc.Wesaythatrationalfunctionsls()ls()12ws()==,ws()(2.1.2)12ms()ms()12belongtothesameequivalenceclassifandonlyiflsmslsms()()=()().(2.1.3)1221Letl1(s)=a(s)l1(s)andm1(s)=a(s)m1(s),wherea
3、(s)isagreatestcommondivisorofl1(s)andm1(s).Thenasls()()11ls()ws()==,1asmsms()()()11108PolynomialandRationalMatriceswherel1(s)andm1(s)arerelativelyprime.Thustherationalfunction(2.1.1)representsthewholeequivalenceclass.Wesaythattherationalfunction(2.1.1)isofstandardformifandonlyifthepolynomialsl(s
4、)andm(s)arerelativelyprimeandthepolynomialm(s)ismonic(i.e.,apolynomialinwhichthecoefficientatthehighestpowerofthevariablesis1).Zerosofthenumeratorpolynomiall(s)arecalledfinitezeros(shortlyzeros),andzerosofthedenominatorpolynomialm(s)arecalledfinitepoles(shortlypoles)oftherationalfunction(2.1.1).
5、Definition2.1.1.Anorderroftherationalfunction(2.1.1)isadifferenceofdegreesofdenominatorm(s)andnumeratorl(s).rm=-deg()deg()sls,(2.1.5)wheredegdenotesthedegreeofapolynomial.Letmm--11nnlslsls()=++...++lslm,()ssas=++...++asa.(2.1.6)mm--110n110Ifthepolynomials(2.1.6)areanumeratoranddenominator,respec
6、tively,ofthefunction(2.1.1),thentheorderofthisfunctionisequaltor=n-m.Thisfunctionhasmfinitezerosandnfinitepoles.Ifr=n-m<0,thenthisfunctionhasapoleofmultiplicityr(s=Ñ)atinfinityandifr=n-m>0,thenthisfunctionhasazeroofmultiplicityr(s=Ñ)atinfinity.Definition2.1.2.Therationalfunction(2.1.1)iscalledpr
7、oper(orcausal)ifandonlyifitsorderisnonnegative(r=degm(s)–degl(s)í0),andstrictlyproper(orstrictlycausal)ifandonlyifitsorderispositive(r=degm(s)–degl(s)>0).Dividingthenumeratorl(s)bythedenominatorm(s),therationalfunction(2.1.1
