Pseudoinverse Formulation of Rayleigh-Schrödinger Perturbation Theory for the Symmetric Definite Generalized Eigenvalue Problem

In this paper, a comprehensive treatment of Rayleigh-Schrödinger perturbation theory for the symmetric definite generalized eigenvalue problem [1, 2] is fur-nished with emphasis on the degenerate problem. The treatment is simply based upon the Moore-Penrose pseudoinverse thus constituting the natural generalization of the procedure for the standard symmetric eigenvalue problem [3]. In addition to providing a concise matrix-theoretic formulation of this procedure, it also pro-vides for the explicit determination of that stage of the algorithm where each higher order eigenvector correction becomes fully determined. Along the way, we gen-eralize the Dalgarno-Stewart identities [15] from the standard to the generalized eigenvalue problem. The general procedure is illustrated by an extensive example