Low-Rank Perturbation of Regular Matrix Pencils with Symmetry Structures

Abstract

The generic change of the Weierstraß canonical form of regular complex structured matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different 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 different behavior for the eigenvalues 0 and \(\infty \), respectively, \(+1\) and \(-1\). The differences arise in those cases where the parity of the partial multiplicities in the perturbed matrix 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 Batzke (Linear Algebra Appl 458:638–670, 2014, Oper Matrices 10:83–112, 2016) 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-1 matrix pencils, as those allow a parametrization of the set of matrix pencils with a given symmetry structure and a given rank.