## 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.

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.

Original language | English |
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Pages (from-to) | 328-364 |

Journal | Linear Algebra and its Applications |

Volume | 511 |

Issue number | 0 |

Early online date | 14 Sept 2016 |

DOIs | |

Publication status | Published - 15 Dec 2016 |