标题： 具有有界秩和固定奇数次数的一般对称矩阵多项式 显示英文标题

标题： Generic symmetric matrix polynomials with bounded rank and fixed odd grade

摘要： 我们确定了奇数阶$d$的$n \times n$复对称矩阵多项式的通用完全特征结构，其秩至多为$r$。 更准确地说，我们证明了奇数阶$d$的$n \times n$复数对称矩阵多项式的集合，即次数至多为$d$，且秩至多为$r$的集合，是具有某些显式描述的完整特征结构的对称矩阵多项式$\lfloor rd/2\rfloor+1$个集合的闭包的并集。 Then, we prove that these sets are open in the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure, in the case where all their elementary divisors are different from each other and of degree $1$. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than one, because the symmetric ones include eigenvalues while the others not. This difference requires to use new techniques. 显示英文摘要

摘要： We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and rank at most $r$ is the union of the closures of the $\lfloor rd/2\rfloor+1$ sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then, we prove that these sets are open in the set of $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure, in the case where all their elementary divisors are different from each other and of degree $1$. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than one, because the symmetric ones include eigenvalues while the others not. This difference requires to use new techniques.