Spectral properties of rational matrix functions with nonnegative realizations

✍ Scribed by K.-H. Förster; B. Nagy

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 742 KB
• Volume: 275-276
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(97)10034-9

✦ Synopsis

If (A, B, C) is an (entrywise) nonnegative realization of a rational matrix function W (i.e. W(I) = C(1 -A))'B for 1.6 o(A)) vanishing at infinity, then Y(W) := inf{r 2 0: W has no poles i, with r < [Ai} is a pole of Wand r(A) := spectral radius of A is an eigenvalue of A. We prove that, if the realization is minimal-nonnegative, then 1. ?-(W) = r(A), 2. order of the pole I-(W) of W = order of the pole r(A) of (. -A)-'. 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.