Extending the notions of companion and infinite companion to matrix polynomials

Abstract

The notion of infinite companion matrix is extended to the case of matrix polynomials (including polynomials with singular leading coefficient). For row reduced polynomials a finite companion is introduced as the compression of the shift matrix. The methods are based on ideas of dilation theory. Connections with systems theory are indicated. Applications to the problem of linearization of matrix polynomials, solution of systems of difference and differential equations and new factorization formulae for infinite block Hankel matrices having finite rank are shown. As a consequence, any system of linear difference or differential equations with constant coefficients can be transformed into a first order system of dimension n = deg det D.