# Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

## TL;DR

This paper characterizes the closure of the set of skew-symmetric matrix polynomials with fixed odd grade and rank, describing their complete eigenstructure and including the special case of skew-symmetric pencils.

## Contribution

It provides an explicit description of the eigenstructure for skew-symmetric matrix polynomials with fixed rank and odd grade, extending to pencils.

## Key findings

- Set of such polynomials is the closure of a set with a complete eigenstructure.
- Eigenstructure description applies to skew-symmetric pencils as a special case.
- Results facilitate understanding of eigenstructure in structured polynomial matrix problems.

## Abstract

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$ and rank at most $2r$. In particular, this result includes the case of skew-symmetric matrix pencils ($d=1$).