Generic matrix polynomials with fixed rank and fixed degree

Abstract

The set ${\mathcal{P}}_{r,d}^{m\times n}$ 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 ${\mathcal{P}}_{r,d}^{m\times n}$ 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 $m\ne n$, we show that ${\mathcal{P}}_{r,d}^{m\times n}$ 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~$r$.

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