The calculation of the distance to a nearby defective matrix

TL;DR

This paper introduces a new fast algorithm to compute the distance from a matrix to the nearest defective matrix, extending existing methods and demonstrating improved performance through numerical examples.

Contribution

It presents a novel, efficient algorithm based on the Implicit Determinant Method for this specific matrix distance problem.

Findings

The algorithm accurately computes the distance to a nearby defective matrix.

Numerical examples show the algorithm's high efficiency and reliability.

Performance benchmarks indicate improvements over previous methods.

Abstract

In this paper a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam & Bora (Linear Algebra Appl., 396 (2005), pp.~273--301) and reduces to finding when a parameter-dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the Implicit Determinant Method introduced by Spence & Poulton (J. Comput. Phys., 204 (2005), pp.~65--81). Numerical results for several examples illustrate the performance of the algorithm.