Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

TL;DR

This paper introduces a Newton-based method for computing multiple eigenvalues and generalized eigenvectors of parameter-dependent matrices, leveraging versal deformation theory, with practical MATLAB implementation and numerical examples.

Contribution

It presents a novel Newton method for finding multiple eigenvalues and generalized eigenvectors in parameter-dependent matrices, extending to the entire matrix space.

Findings

Method successfully computes multiple eigenvalues with given multiplicity.

Applicable to matrices with or without parameters.

MATLAB implementation demonstrates practical utility.

Abstract

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. The implementation of the method in MATLAB code is available.