Generic rank-one perturbations of structured regular matrix pencils

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 of size one under a rank-1 perturbation, depending on that eigenvalue occurring 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 shrunk 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.