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Preprint 03-2014

Generic rank-one perturbations of structured regular matrix pencils

Source file is available as :   Portable Document Format (PDF)

Author(s) : Leonhard Batzke

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 03-2014

MSC 2000

15A22 Matrix pencils
47A55 Perturbation theory

Abstract :
Classes of regular, structured matrix pencils are examined with respect to their spectral behavior under a certain type of structure-preserving rank-1 perturbations. The observed effects are as follows: On the one hand, generically the largest Jordan block at each eigenvalue gets destroyed or becomes size one under a rank-1 perturbation, depending on that eigenvalue occuring in the perturbating pencil or not. On the other hand, certain Jordan blocks of T-alternating matrix pencils occur in pairs, so that in some cases, the largest block cannot just be destroyed or shrinked to size one without violating the pairing. Thus, the largest remaining Jordan block will typically increase in size by one in these cases. Finally, these results are shown to carry over to the classes of palindromic and symmetric matrix pencils.

Keywords : Matrix pencil, alternating matrix pencil, palindromic matrix pencil, symmetric matrix pencil, perturbation theory, rank one perturbation, generic perturbation.

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