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

In this paper, a comprehensive treatment of analytic Rayleigh-Schrödinger perturbation theory for the symmetric definite generalized eigenvalue problem [1, 2] is furnished 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 linear perturbation of the standard symmetric eigenvalue problem [3, 4, 5]. 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. Along the way, we generalize the Dalgarno-Stewart identities [18] from linear perturbation of the standard symmetric eigenvalue problem to analytic perturbation of the symmetric definite generalized eigenvalue problem. An extensive example illustrates the general procedure.