Spectral properties of rational matrix functions with nonnegative realizations

AbstractIf (A,B,C) is an (entrywise) nonnegative realization of a rational matrix function W (i.e. W(λ) = C(λ − A)−1B for λ ∉ σ(A)) vanishing at infinity, then r(W) := inf{r ⩾ 0: W has no poles λ with r < |λ|} is a pole of W and r(A) := spectral radius of A is an eigenvalue of A. We prove that, if the realization is minimal-nonnegative, then 1.1. r(W) = r(A),2.2. order of the pole r(W) of W = order of the pole r(A) of (· − A)−1.We characterize the order of these poles in the spirit of Rothblum's index theorem, namely as the length of the longest chains of singular vertices in the reduced graph of A with a suitable new access relation, which incorporates B and C into the familiar access relation of A