On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block

Abstract

Let A(z) be an analytic square matrix and $\lambda_{0}$ an eigenvalue of A(0) of multiplicity m. Then under the generic condition, the characteristic polynomial of A(z) evaluated at $\lambda_{0}$ has a simple zero at z=0, we prove that the Jordan normal form of A(0) corresponding to the eigenvalue $\lambda_{0}$ consists of a single m-by-m Jordan block, the perturbed eigenvalues near $\lambda_{0}$ and their eigenvectors can be represented by a single convergent Puiseux series containing only powers of z^{1/m}, and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of A(z) 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.