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Main Title: On the sign characteristics of Hermitian matrix polynomials
Author(s): Mehrmann, Volker
Noferini, Vanni
Tisseur, Françoise
Xu, Hongguo
Type: Article
Language Code: en
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.
Issue Date: 14-Sep-2016
Date Available: 14-Nov-2019
DDC Class: 510 Mathematik
Subject(s): hermitian matrix polynomial
sign characteristic
sign characteristic at infinity
sign feature
signature constraint
perturbation theory
Sponsor/Funder: EC/FP7/EU/267526/Functions of Matrices: Theory and Computation/MATFUN
DFG, 5485610, FZT 86: Matheon - Mathematik für Schlüsseltechnologien: Modellierung, Simulation und Optimierung realer Prozesse
Journal Title: Linear Algebra and its Applications
Publisher: Elsevier
Publisher Place: Amsterdam
Volume: 511
Publisher DOI: 10.1016/j.laa.2016.09.002
Page Start: 328
Page End: 364
EISSN: 1873-1856
ISSN: 0024-3795
Appears in Collections:FG Numerische Mathematik » Publications

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