Generic rank-two perturbations of structured regular matrix pencils

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Author(s) : Leonhard Batzke

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 09-2014

MSC 2000

15A22 Matrix pencils
47A55 Perturbation theory

Abstract :
The spectral behavior of classes of structured regular matrix pencils is examined under certain structure-preserving rank-2 perturbations. For T-alternating, palindromic, and skew-symmetric matrix pencils we observe the following effects at each eigenvalue $\lambda$ under a generic, structure-preserving rank-2 perturbation: 1) The largest two Jordan blocks at $\lambda$ are destroyed. 2) If hereby the eigenvalue pairing imposed by the structure is violated, also the largest remaining Jordan block at $\lambda$ will grow in size by one. 3) If $\lambda$ is a single (double) eigenvalue of the perturbating pencil, one (two) new Jordan blocks of size one will be created at $\lambda$.

Keywords : Matrix pencil, alternating matrix pencil, palindromic matrix pencil, skew-symmetric matrix pencil, perturbation theory, rank two perturbation, generic perturbation