Preserving spectral properties of structured matrices under structured perturbations

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.

Eigenvalue embedding problem for quadratic regular matrix polynomials with symmetry structures

“…On the other hand, as invariant subspaces of matrices can be interpreted as a generalization of eigenvectors, deflating pairs and invariant pairs can be seen as generalizations of eigenpairs for matrix pencils and matrix polynomials respectively. Recently, structure-preserving perturbations for structured matrices and matrix pencils are determined to preserve invariant subspaces and deflating pairs respectively in [26,1]. Then these results are used to reproduce a desired scalars as eigenvalues of the perturbed structured matrices and matrix pencils without effecting a set of desired eigenpairs of the unperturbed matrix and matrix pencils, respectively.…”