Generic symmetric matrix polynomials with bounded rank and fixed odd grade

TL;DR

This paper characterizes the generic eigenstructures of complex symmetric matrix polynomials with odd grade and bounded rank, revealing structures that include eigenvalues, unlike previous results for other matrix polynomial classes.

Contribution

It provides a complete description of the generic eigenstructures for symmetric matrix polynomials of odd grade and bounded rank, including explicit conditions and open sets.

Findings

Identifies the union of closures of specific eigenstructure sets as generic

Proves these sets are open in the space of symmetric matrix polynomials

Highlights the difference in eigenstructure inclusion compared to other polynomial classes

Abstract

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.