Exploring defective eigenvalue problems with the method of lifting
Abstract
Consider an N x N matrix A for which zero is a defective eigenvalue. In this case, the algebraic multiplicity of the zero eigenvalue is greater than the geometric multiplicity. We show how an inflated (N+1) x (N+1) matrix L can be constructed as a rank one perturbation to A, such that L is singular but no longer defective, and the nullvectors of L can be easily related to the nullvectors of A. The motivation for this construction comes from linear stability analysis of an experimental reaction-diffusion system which exhibits the Turing instability. The utility of this scheme is accurate numerical computation of nullvector(s) corresponding to a defective zero eigenvalue. We show that numerical computations on L yield more accurate eigenvectors than direct computation on A.
Cite
@article{arxiv.math-ph/0210063,
title = {Exploring defective eigenvalue problems with the method of lifting},
author = {S. Setayeshgar and H. B. Keller and J. E. Pearson},
journal= {arXiv preprint arXiv:math-ph/0210063},
year = {2007}
}
Comments
16 pages, including 4 figures