A Perturbation Method for Index Detection for Linear Matrix Pencils

TL;DR

This paper develops rigorous bounds and error analysis for eigenvalue expansions at infinity in linear matrix pencils, estimates eigenvector condition numbers under perturbations, and validates findings through numerical simulations.

Contribution

It introduces a perturbation method for index detection in linear matrix pencils, providing non-asymptotic bounds and applying the results to the Cayley transform.

Findings

Non-asymptotic bounds for Puiseux expansion at infinity

Estimated expected eigenvector condition number under random perturbations

Numerical simulations confirming theoretical results

Abstract

Rigorous, non-asymptotic bounds for the Puiseux expansion of the eigenvalue at infinity are given. Error analysis is provided. Further, the expected value of the eigenvector condition number of a randomly perturbed matrix is estimated. The latter result is applied to the Cayley transform of the linear pencil. Numerical simulations illustrating the theoretical findings are provided.