# First-order Perturbation Theory for Eigenvalues and Eigenvectors

## TL;DR

This paper develops and compares two methods for first-order perturbation analysis of eigenvalues and eigenvectors of general square matrices, extending classical results and providing practical verification techniques.

## Contribution

It introduces a novel block-diagonalization proof for eigenvector perturbation and discusses extensions, normalization, and computational verification of the theory.

## Key findings

- Two distinct proofs of eigenvector perturbation theorem presented
- Extension of perturbation theory to various normalizations
- Guidance on computational verification of results

## Abstract

We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is not so well known outside a community of specialists. We give two different proofs of the main eigenvector perturbation theorem. The first, a block-diagonalization technique inspired by the numerical linear algebra research community and based on the implicit function theorem, has apparently not appeared in the literature in this form. The second, based on complex function theory and on eigenprojectors, as is standard in analytic perturbation theory, is a simplified version of well-known results in the literature. The second derivation uses a convenient normalization of the right and left eigenvectors defined in terms of the associated eigenprojector, but although this dates back to the 1950s, it is rarely discussed in the literature. We then show how the eigenvector perturbation theory is easily extended to handle other normalizations that are often used in practice. We also explain how to verify the perturbation results computationally. We conclude with some remarks about difficulties introduced by multiple eigenvalues and give references to work on perturbation of invariant subspaces corresponding to multiple or clustered eigenvalues. Throughout the paper we give extensive bibliographic commentary and references for further reading.