Preprint 32-2015

On the sign characteristics of Hermitian matrix polynomials

Author(s) : Volker Mehrmann , Vanni Noferini , Francoise Tisseur , Hongguo Xu

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 32-2015

MSC 2000

15A57 Other types of matrices

65F15 Eigenvalues, eigenvectors

Abstract :
The sign characteristics of Hermitian matrix polynomials are discussed, and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity. The concept of sign characteristic arises in different forms in many scientific fields, and is essential for the stability analysis in Hamiltonian systems or the perturbation behavior of eigenvalues under structured perturbations. We extend classical results by Gohberg, Lancaster, and Rodman to the case of infinite eigenvalues. We derive a systematic approach, studying how sign characteristics behave after an analytic change of variables, including the important special case of Möbius transformations, and we prove a signature constraint theorem. We also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree Hermitian matrix polynomials, while it may change for odd degree matrix polynomials. We argue that the non-uniformity can be resolved by introducing an extra zero leading matrix coefficient.

Keywords : Hermitian matrix polynomial, sign characteristic, sign characteristic at infinity, sign feature, signature constraint, perturbation theory