Even grade generic skew-symmetric matrix polynomials with bounded rank

Abstract

We show that the set of $m\times m$ complex skew-symmetric matrix polynomials of even 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 certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic $m\times m$ complex skew-symmetric matrix polynomials of even grade d and rank at most 2r. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [24].