Generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree

Abstract

The set POLd,rmxn of m x 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 POLd,rmxn 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 = n} and m not equal n, we show that POLd,rmxn 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 x n matrix polynomials of grade d and rank at most r.