Quasi commutativity of regular matrix polynomials: Resultant and Bezoutian

Abstract

In a recent paper of I. Gohberg and the authors necessary and sufficient conditions are obtained in order that for two regular matrix polynomials L and M the dimension of the null space of the associate square resultant matrix is equal to the sum of the multiplicities of the common zeros of L and M, infinity included. The conditions are stated in terms of quasi commutativity. In the case of commuting matrix polynomials, in particular, in the scalar case, these conditions are automatically fulfilled. The proofs in the above paper are heavily based on the spectral theory of matrix polynomials. In the present paper a new proof is given of the sufficiency part of the result mentioned above. Here we use the connections between the Bezout and resultant matrices and a general abstract scheme for determining the null space of the Bezoutian of matrix polynomials which is based on a state space analysis of Bezoutians.