Constructive Perturbation Theory for Matrices with Degenerate Eigenvalues

Abstract. Let A (ε) be an analytic square matrix and λ0 an eigenvalue of A (0) of multiplicity m ≥ 1. Then under the generic condition, ∂ ∂ε det (λI −A (ε)) |(ε,λ)=(0,λ0) 6= 0, we prove that the Jordan normal form of A (0) corresponding to the eigenvalue λ0 consists of a single m × m Jordan block, the perturbed eigenvalues near λ0 and their eigenvectors can be represented by a single convergent Puiseux series containing only powers of ε1/m, and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of A (ε) at the origin. Using these recursive formulas we calculate the series coefficients up to the second order and list them for quick reference. This paper gives, under a generic condition, explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors for non-selfadjoint analytic perturbations of matrices with degenerate eigenvalues.