On nonmonic quadratic matrix polynomials with nonnegative coefficients

The matrix polynomial Q(lambda) = lambda I-S(lambda), 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 - rho(S(r)) (here g(.) denotes the spectral radius). The existence of a nonnegative (spectral) matrix root of Q(.) will be related to the existence of a positive r > g(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 = rho(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 rho(S(.)) and on its derivatives.