Sensitivity and computation of a defective eigenvalue

TL;DR

This paper investigates the sensitivity of defective eigenvalues to data perturbations and introduces a regularized least squares approach for their accurate numerical computation despite perturbations.

Contribution

It establishes a bounded sensitivity theory for defective eigenvalues under specific perturbations and proposes a regularized computational method for improved numerical stability.

Findings

Sensitivity of defective eigenvalues can be finitely bounded under certain perturbations.

A least squares formulation enables accurate computation of defective eigenvalues.

The method remains stable even with approximate data and round-off errors.

Abstract

A defective eigenvalue is well documented to be hypersensitive to data perturbations and round-off? errors, making it a formidable challenge in numerical computation particularly when the matrix is known through approximate data. This paper establishes a finitely bounded sensitivity of a defective eigenvalue with respect to perturbations that preserve the geometric multiplicity and the smallest Jordan block size. Based on this perturbation theory, numerical computation of a defective eigenvalue is regularized as a well-posed least squares problem so that it can be accurately carried out using floating point arithmetic even if the matrix is perturbed.