Generic matrix polynomials with fixed rank and fixed degree

2016/12/13 by Andrii Dmytryshyn, Froilán M. Dopico, Dmytryshyn, Andrii +1
Computer Science · Mathematics · #15A18 #15A21 #Advanced Topics in Algebra #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1612.04085

openalex publication_date 2016/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The set \cal P^{m× n}_r,d of m × 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, … , min \m, n\-1, we show that \cal P^{m× 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 ≠ n, we show that \cal P^{m× 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 × n matrix polynomials of grade d and rank at most~r.

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