Eigenvalues of the real generalized eigenvalue equation perturbed by a low-rank perturbation

Abstract

The low-rank perturbation (LRP) method solves the perturbed eigenvalue equation (B +V)Ψ _k = ɛ _k (C +P)Ψ _k , where the eigenvalues and the eigenstates of the related unperturbed eigenvalue equationBΦ _i = λ _i CΦ _i are known. The method is designed for arbitraryn-by-n matricesB, V, C, andP, with the only restriction that the eigenstates Φ _i of the unperturbed equation should form a complete set. We consider here a real LRP problem where all matrices are Hermitian, and where in addition matricesC and (C +P) are positive definite. These conditions guarantee reality of the eigenvalues ɛ _k and λ _i . In the original formulation of the LRP method, each eigenvalue ɛ _k is obtained iteratively, starting from some approximate eigenvalue ɛ ^′ _k . If this approximate eigenvalue is not well chosen, the iteration may sometimes diverge. It is shown that in the case of a real LRP problem, this danger can be completely eliminated. If the rank ρ of the generalized perturbation {V, P} is “small” with respect ton, then one can easily bracket and hence locate to any desirable accuracy the eigenvalues ɛ _k (k = 1, ...,n) of the perturbed equation. The calculation of alln eigenvalues requiresO(ρ² n ²) operations. In addition, if the perturbation (V, P) is local with the localizabilityl ≈p, then onlyO(ρ² n) operations are required for a derivation of a single eigenvalue.