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## Sign characteristics of regular Hermitian matrix pencils under generic rank-1 and rank-2 perturbations

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Author(s) : Leonhard Batzke

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 36-2014

MSC 2000

15A22 Matrix pencils
47A55 Perturbation theory

Abstract :
The spectral behavior of regular Hermitian matrix pencils is examined under certain structure-preserving rank-1 and rank-2 perturbations. Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form, it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. The observed effects are as follows: Under a rank-1 or rank-2 perturbation, generically the largest one or two, respectively, Jordan blocks at each eigenvalue $\lambda$ are destroyed, and if $\lambda$ is an eigenvalue of the perturbation, also one new block of size one is created at $\lambda$. If $\lambda$ is real (or infinite), additionally all signs at $\lambda$ but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. Also, if the potential new block of size one is real, its sign is in most cases prescribed to be the sign that is attached to the eigenvalue $\lambda$ in the perturbation.

Keywords : Matrix pencil, Hermitian matrix pencil, sign characteristic, rank one perturbation, rank two perturbation, generic perturbation