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

TL;DR

This paper derives explicit recursive formulas to compute eigenvalues and eigenvectors of perturbed matrices with Jordan blocks, using Puiseux series expansions and derivatives of the matrix at the origin.

Contribution

It introduces new recursive formulas for calculating perturbed eigenvalues and eigenvectors in Jordan block cases, enhancing spectral perturbation analysis methods.

Findings

Explicit recursive formulas for Puiseux series coefficients

Calculation of series coefficients up to second order

Representation of perturbed eigenvalues and eigenvectors as convergent Puiseux series

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.