Zero distribution of matrix polynomials

AbstractThe theory of finite dimensional reproducing kernel Krein spaces is exploited to obtain matrix analogues of the Schur–Cohn theorem and the Hermite theorem on the distribution of zeros. Formulas for the number of zeros of the determinant of an m×m matrix polynomial N(λ) inside and outside the region Ω+ in terms of the signature of an associated matrix (that is subsequently identified as a Bezoutian in the sense of Haimovici and Lerer for appropriately chosen realizations of the polynomials under consideration) are developed when detN(λ)≠0 on the boundary of Ω+ and Ω+ is taken equal to either the open unit disk or the open upper half-plane. The proof is reasonably self contained and reasonably uniform for both choices of Ω+. The conditions imposed are less restrictive than those imposed in the papers [H. Dym, N. Young, A Schur–Cohn theorem for matrix polynomials, Proceedings of the Edinburgh Mathematical Society, vol. 33, 1990, pp. 337–366] and [H. Dym, A Hermite theorem for matrix polynomials, in: Operator Theory: Advances and Applications, vol. 50, Birkhäuser Verlag, Basel, 1991, pp. 191–214]. Comparisons with the Anderson–Jury Bezoutian and the Haimovici–Lerer Bezoutians referred to above are made, and an application to block Toeplitz matrices is given