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On the sign characteristics of Hermitian matrix polynomials. (English) Zbl 1352.15027

Summary: 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 I. Gohberg et al. [Ann. Math. (2) 112, 33–71 (1980; Zbl 0446.15007)] 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.

MSC:

15A54 Matrices over function rings in one or more variables
15B57 Hermitian, skew-Hermitian, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion

Citations:

Zbl 0446.15007

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References:

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