Structure-preserving eigenvalue modification of symplectic matrices and matrix pencils

TL;DR

This paper extends eigenvalue modification techniques to symplectic matrices and matrix pencils, enabling structure-preserving spectrum adjustments with bounds and condition number analysis.

Contribution

It generalizes Brauer's and Rado's theorems for symplectic matrices, providing methods for eigenvalue modification that preserve structure and include bounds and condition number insights.

Findings

Eigenvalue modification of symplectic matrices using structure-preserving updates

Universal bounds on the distance between original and modified matrices

Extension of results to matrix pencils

Abstract

A famous theorem by R. Brauer shows how to modify a single eigenvalue of a matrix by a rank-one update without changing the remaining eigenvalues. A generalization of this theorem (due to R. Rado) is used to change a pair of eigenvalues of a symplectic matrix S in a structure-preserving way to desired target values. Universal bounds on the relative distance between S and the newly constructed symplectic matrix S' with modified spectrum are given. The eigenvalues Segre characteristics of S' are related to those of S and a statement on the eigenvalue condition numbers of S' is derived. The main results are extended to matrix pencils.