Spectral properties of rational matrix functions with nonnegative realizations Original Research Article

If (A,B,C) is an (entrywise) nonnegative realization of a rational matrix function W (i.e. W(λ) = C(λ − A)−1B for λ negated set membership σ(A)) vanishing at infinity, then r(W) := inf{r greater-or-equal, slanted 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 Rothblumʹs 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.