Generic matrix polynomials with fixed rank and fixed degree

TL;DR

This paper characterizes the structure and eigenstructures of complex matrix polynomials with fixed rank and degree, revealing their geometric properties and closures in the space of all such polynomials.

Contribution

It provides a detailed description of the closures of sets of matrix polynomials with fixed rank and degree, including explicit eigenstructures and their union representations.

Findings

Sets of fixed-rank matrix polynomials are unions of closures of rank-specific subsets.

Complete eigenstructures are explicitly described for these sets.

Full-rank rectangular polynomials form a single closure of a specific eigenstructure set.

Abstract

The set ${\cal P}^{m\times n}_{r,d}$ of $m \times n$ complex matrix polynomials of grade $d$ and (normal) rank at most $r$ in a complex $(d+1)mn$ dimensional space is studied. For $r = 1, \dots , \min \{m, n\}-1$, we show that ${\cal P}^{m\times n}_{r,d}$ is the union of the closures of the $rd+1$ sets of matrix polynomials with rank $r$, degree exactly $d$, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. $r= \min \{m, n\}$ and $m \neq n$, we show that ${\cal P}^{m\times n}_{r,d}$ coincides with the closure of a single set of the polynomials with rank $r$, degree exactly $d$, and the described complete eigenstructure. These complete eigenstructures correspond to generic $m \times n$ matrix polynomials of grade $d$ and rank at most~$r$.