奇异Hankel矩阵

SINGULAR HANKEL MATRICES

本文研究有限奇异Hankel矩阵的相容多项式与最小度.按定义,非零多项式f(z)=■f_jz^j为n阶Hankel矩阵H=(h_(i+j))■的相容多项式,假如■[f]_n=0,这里[f]_n=(f_0,…,f_n)~■与~■=(h_(i+j))~■每个n阶Hankel矩阵H均可由某个有理函数g/f(degg≤degf)生成:H=H_n(g/f),如果有degf=q。

Finite Hankel matrices are characterized in terms of the Compatible polynomials and the minimal degree.Let H of order n be a singu- lar Hankel matrix with rank H=r>0 and the H-polynomial a(2)of degree m.Then H is proper(resp.degenerate)iff H is compatible with some monic polynomial of degree n(resp.with polynomial 1);or iff the family of all polynomials compatible with H is the set{f 0:degf≤n and a(z)|f(z)} (resp.{f 0:degf≤n-r}).Moreover,H has the minimal degree equalled m if H is proper,and 2 n-r otherwise.