Exploring defective eigenvalue problems with the method of lifting

S. Setayeshgar; H. B. Keller; J. E. Pearson

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Exploring defective eigenvalue problems with the method of lifting

S. Setayeshgar·H. B. Keller·J. E. Pearson

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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.

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S. Setayeshgar, H. B. Keller, J. E. Pearson. 2002-10-31. Exploring defective eigenvalue problems with the method of lifting. https://arxiv.org/abs/math-ph/0210063

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