On the sign characteristics of Hermitian matrix polynomials

Research output: Contribution to journalArticle

  • Authors:
  • Volker Mehrmann
  • Vanni Noferini
  • Francoise Tisseur
  • Hongguo Xu


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

Bibliographical metadata

Original languageEnglish
Early online date14 Sep 2016
StatePublished - 15 Dec 2016