A Hermite Theorem for Matrix Polynomials

Abstract

An analogue of the Hermite theorem for the number of zeros in a half plane for a scalar polynomial is obtained for a class of m × m matrix polynomials by (finite dimensional) reproducing kernel Krein space methods. The paper, which is largely expository, is partially modelled on an earlier paper with N.J. Young which developed similar analogues of the Schur-Cohn theorem for matrix polynomials. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods. New proofs of some recent results on the distribution of the roots of certain matrix polynomials which are associated with invertible Hermitian block Hankel and block Toeplitz matrices are presented as an application of the main theorem.