Journal of Applied Mathematics
Volume 2003 (2003), Issue 9, Pages 459-485
doi:10.1155/S1110757X03303092
Pseudoinverse formulation of Rayleigh-Schrödinger perturbation
theory for the symmetric matrix eigenvalue problem
Applied Mathematics, Kettering University, 1700 West Third Avenue, Flint 48504-4898, MI, USA
Received 20 March 2003
Copyright © 2003 Brian J. McCartin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A comprehensive treatment of Rayleigh-Schrödinger perturbation theory for the symmetric matrix eigenvalue problem is furnished with emphasis on the degenerate problem. The treatment is simply based upon the Moore-Penrose pseudoinverse thus distinguishing it from alternative approaches in the literature. In addition to providing a concise matrix-theoretic formulation of this procedure, it also provides for the explicit determination of that stage of the algorithm where each higher-order eigenvector correction becomes fully determined. The theory is built up gradually with each successive stage appended with an illustrative example.