Preserving spectral properties of structured matrices under structured perturbations

TL;DR

This paper investigates how to modify structured matrices through perturbations to preserve their spectral properties, such as eigenvalues and invariant subspaces, while enabling targeted eigenvalue adjustments.

Contribution

It introduces methods for structure-preserving perturbations that maintain spectral properties and Jordan structures, including no spillover perturbations with minimal rank impact.

Findings

Structured perturbations can preserve eigenvalues and invariant subspaces.

Methods enable modification of specific eigenvalues without affecting others.

No spillover perturbations can alter selected eigenvalues with minimal rank.

Abstract

This paper is devoted to the study of preservation of eigenvalues, Jordan structure and complementary invariant subspaces of structured matrices under structured perturbations. Perturbations and structure-preserving perturbations are determined such that a perturbed matrix reproduces a given subspace as an invariant subspace and preserves a pair of complementary invariant subspaces of the unperturbed matrix. These results are further utilized to obtain structure-preserving perturbations which modify certain eigenvalues of a given structured matrix and reproduce a set of desired eigenvalues while keeping the Jordan chains unchanged. Moreover, a no spillover structured perturbation of a structured matrix is obtained whose rank is equal to the number of eigenvalues (including multiplicities) which are modified, and in addition, preserves the rest of the eigenvalues and the corresponding Jordan chains which need not be known. The specific structured matrices considered in this paper form Jordan and Lie algebra corresponding to an orthosymmetric scalar product.