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

Let A(ε) be an analytic square matrix and λ₀ an eigenvalue of $A(0)$ of algebraic multiplicity m≥1. Then under the condition ∂/∂ε(λ I-A(ε))|_(ε,λ)=(0,λ₀)≠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 corresponding 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-self-adjoint analytic perturbations of matrices with nonderogatory eigenvalues.