SPECTRAL PROPERTIES OF RATIONAL MATRIX FUNCTIONS WITH NONNEGATIVE REALIZATIONS

Abstract If ( A,B,C ) is an (entrywise) nonnegative realization of a rational matrix function W (i.e. W ( λ ) = C ( λ − A ) −1 B 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 .