Preprint 04-2019

Low rank perturbation of regular matrix pencils with symmetry structures

Author(s) : Fernando De Teran, Christian Mehl, Volker Mehrmann

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 04-2019

MSC 2000

15A22 Matrix pencils

15A18 Eigenvalues, singular values, and eigenvectors

Abstract :
The generic change of theWeierstra Canonical Form of regular complex structured matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several dierent symmetry structures are considered and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a dierent behavior for the eigenvalues 0 and 1, respectively +1 and 􀀀1. The dierences arise in those cases where the parity of the partial multiplicities in the perturbed pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-1 and rank- 2 cases that were obtained in [3, 5] for the case of special structure-preserving perturbations. As the main tool, we use decompositions of matrix pencils with symmetry structure into sums of rank-one pencils, as those allow a parametrization of the set of matrix pencils with a given symmetry structure and a given rank.

Keywords : Even matrix pencil, palindromic matrix pencil, Hermitian matrix pencil, symmetric matrix pencil, skew-symmetric matrix pencil, perturbation analysis, generic perturbation, low-rank perturbation, additive decomposition of structured pencils, Weierstrass canonical form