Session B7 - Numerical Linear Algebra
July 15, 17:00 ~ 17:30 - Room B5
Parameter-Dependent Rank-One Perturbations of Singular Hermitian Pencils
TU Berlin, Germany - email@example.com
We investigate the effect of perturbations of singular Hermitian pencils $\lambda E+A$ that are (1) of rank one, (2) structure-preserving, (3) generic, (4) parameter-dependent, and (5) regularizing, i.e., generic perturbations that have the form \[ \lambda (E+\tau euu^T)+A+\tau auu^T, \] where the perturbed pencil is regular.
It is known that in this situation the eigenvalues of the perturbed pencil are constant in the parameter $\tau$. One would now expect that for a generic regularizing rank-one perturbation all eigenvalues of the perturbed pencil are simple. While this is indeed the case for pencils without symmetry structure, surprisingly this is different for real symmetric pencils where it happens that generically double eigenvalues will occur. While these eigenvalues will be constant in $\tau$ as expected, this will not be the case for their geometric multiplicities.
In this talk, we will explain why this peculiar behavior occurs and investigate what happens when the geometric multiplicity of eigenvalues changes.
Joint work with Volker Mehrmann (TU Berlin) and Michal Wojtylak (University of Krakow).