On Nonmonic Quadratic Matrix Polynomials with Nonnegative Coefficients

Abstract

The matrix polynomial Q(λ) = λI-S(λ), where S(·) is a nonmonic quadratic matrix polynomial with (entrywise) nonnegative square matrix coefficients, will be studied. We describe the distribution of the eigenvalues of Q(·), depending on the sign of function r↦ r - ϱ(S(r)) (here ϱ(·) denotes the spectral radius). The existence of a nonnegative (spectral) matrix root of Q(·) will be related to the existence of a positive r > ϱ(S(r)). Assuming that S(t) is irreducible for one positive t, we describe the spectrum of Q(·) on the circles with radius r for any r = ϱ(S(r)) > 0, and describe the possibilities for the existence of a nonnegative matrix root of Q(·), for the properties of a corresponding M-matrix and the spectral properties of Q(·), depending on the function ϱ(S(·)) and on its derivatives.