# Low rank perturbation of regular matrix pencils with symmetry structures

## TL;DR

This paper investigates how low-rank, structure-preserving perturbations affect the eigenvalues and canonical forms of regular matrix pencils with various symmetry structures, revealing both generic and structure-specific behaviors.

## Contribution

It extends existing results by analyzing the impact of low-rank perturbations on structured matrix pencils, especially for complex symmetry types, and introduces a decomposition method for these pencils.

## Key findings

- Eigenvalue changes are generally similar to unstructured cases for most symmetries.
- Special behaviors occur for eigenvalues 0, ∞, +1, -1 in certain symmetry structures.
- Decomposition into rank-one pencils aids in parametrizing structured pencils with fixed rank.

## 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 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 [L. Batzke, Generic Low-Rank Perturbations of Structured Regular Matrix Pencils and Structured Matrices, PhD Thesis, TU Berlin, Berlin, Germany, 2015] and [L. Batzke, Generic rank-two perturbations of structured regular matrix pencils, 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-one pencils, as those allow a parametrization of the set of matrix pencils with a given symmetry structure and a given rank.