Spectral Bounds for Matrix Polynomials with Unitary Coefficients

Abstract

It is well known that the eigenvalues of any unitary matrix lie on the unit circle. The purpose of this paper is to prove that the eigenvalues of any matrix polynomial, with unitary coefficients, lie inside the annulus A_{1/2,2) := {z ∈ C | 1/2 < |z| < 2}. The foundations of this result rely on an operator version of Rouche’s theorem and the intermediate value theorem.