On the sign characteristics of Hermitian matrix polynomials

Volker Mehrmann, Vanni Noferini, Francoise Tisseur, Hongguo Xu

    Research output: Contribution to journalArticlepeer-review


    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
    Original languageEnglish
    Pages (from-to)328-364
    JournalLinear Algebra and its Applications
    Issue number0
    Early online date14 Sept 2016
    Publication statusPublished - 15 Dec 2016


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