Exploring defective eigenvalue problems with the method of lifting

Details Exploring defective eigenvalue problems with the method of lifting Exploring defective eigenvalue problems with the method of lifting

2002-10-31

arxiv.org/abs/math-ph/0210063v1

Physics

Mathematical Physics

16 pages, including 4 figures

Scientific paper

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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