资源描述：
《polynomial matrices》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、1PolynomialMatrices1.1PolynomialsLettingbeafield,e.g.,oftherealnumbers,thecomplexnumbers,therationalnumbers,therationalfunctionsW(s)ofacomplexvariables,etc.,ninws()==ƒasina01+as+...+as(1.1.1)i=0iscalledapolynomialw(s)inthevariablesoverthefield,whereai≠for
2、i=0,1,...,narecalledthecoefficientsofthispolynomial.Thesetofpolynomials(1.1.1)overthefieldwillbedenotedby[s].Ifanò0,thenthenonnegativeintegralniscalledthedegreeofapolynomialandisdenoteddegw(s),i.e.,n=degw(s).Thepolynomial(1.1.1)iscalledmonic,ifan=1andzero
3、polynomial,ifai=0fori=0,1,…,n.Thesumoftwopolynomialsnwsaas()=+++...as,(1.1.2a)101nmwsbbs()=+++...bs,(1.1.2b)201misdefinedinthefollowingwaymn½ii°°ƒƒ()absii++asnmi,>°°ii==01m+°°niwsws12()+=()®¾ƒ(absnmii+=),.(1.1.3)°°i=0°°nmii°°ƒƒ()absii++bis,mn>¯¿ii==01n+2P
4、olynomialandRationalMatricesIfn>m,thenthesumisapolynomialofdegreen,ifm>nthenthesumisapolynomialofdegreem.Ifn=mandan+bnò0,thenthissumisapolynomialofdegreenandapolynomialofdegreelessthann,ifan+bn=0.Thuswehavedeg[wsws12()+Ç()]maxdeg»º¬¼[ws1(),deg][ws2()].(1.
5、1.4)Inthesameveinwedefinethedifferenceoftwopolynomials.Apolynomialwhosecoefficientsaretheproductsofthecoefficientsaiandthescalarl,i.e.,nillw()s=ƒasi,(1.1.5)i=0iscalledtheproductofthepolynomial(1.1.1)andthescalarl(ascalarcanberegardedasapolynomialofzerodeg
6、ree).Apolynomialoftheformnm+iwsws12()()=ƒcsi(1.1.6a)i=0iscalledtheproductofthepolynomials(1.1.2),whereicaik==ƒbii-k,0,1,>,n+mk=0(1.1.6b)(ak=>=>0forn,bk0form).kkFrom(1.1.6a)itfollowsthatdeg[wsws12()()]=+nm,(1.1.7)sinceanbmò0foranò0,bmò0.Letw2(s)in(1.1.2)be
7、anonzeropolynomialandn>m,thenthereexistexactlytwopolynomialsq(s)andr(s)suchthatwswsqsrs()=+()()(),(1.1.8)12wheredeg[rs()]<=deg[ws2()]m.(1.1.9)Thepolynomialq(s)iscalledtheintegerpartwhenr(s)ò0andthequotientwhenr(s)=0,andr(s)iscalledtheremainder.PolynomialM
8、atrices3Ifr(s)=0,thenw1(s)=w2(s)q(s);wesaythenthatpolynomialw1(s)isdivisiblewithoutremainderbythepolynomialw2(s),orequivalently,thatpolynomialw2(s)divideswithoutremainderapolynomialw1(s),whichisdenotedbyw1(s)
9、w2(s).
