Skew-symmetric matrix polynomials and their Smith forms

Abstract

Two canonical forms for skew-symmetric matrix polynomials over arbitrary fields are characterized—the Smith form, and its skew-symmetric variant obtained via unimodular congruences. Applications include the analysis of the eigenvalue and elementary divisor structure of products of two skew-symmetric matrices, the derivation of a Smith-McMillan-like canonical form for skew-symmetric rational matrices, and the construction of minimal symmetric factorizations of skew-symmetric rational matrices. A sufficient condition for the existence of solutions to matrix polynomial Sylvester equations, and results on the existence and construction of structured linearizations for regular and singular skew-symmetric matrix polynomials are also presented.