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On the sign characteristics of Hermitian matrix polynomials

Mehrmann, Volker and Noferini, Vanni and Tisseur, Françoise and Xu, Hongguo (2016) 'On the sign characteristics of Hermitian matrix polynomials.' Linear Algebra and its Applications, 511. 328 - 364. ISSN 0024-3795

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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.

Item Type: Article
Uncontrolled Keywords: Hermitian matrix polynomial, Sign characteristic, Sign characteristic at infinity, Sign feature, Signature constraint, Perturbation theory
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Jim Jamieson
Date Deposited: 31 Oct 2016 10:23
Last Modified: 30 Jul 2018 10:15

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