A brief overview of spectral perturbation Theory

TL;DR

This paper provides a concise overview of spectral perturbation theory, focusing on bounds for eigenvalues, eigenvectors, and invariant subspaces across matrices, operators, and holomorphic functions, with simplified proofs and comprehensive analysis.

Contribution

It offers a unified, simplified presentation of spectral perturbation results for matrices, operators, and holomorphic functions, including detailed analysis of invariant subspaces and spectral projections.

Findings

Bounds for perturbed eigenvalues and eigenvectors

Analysis of invariant subspaces under perturbation

Discussion of spectral projections and analytic perturbation effects

Abstract

The aim of this article is to present a brief overview of spectral perturbation theory for matrices, bounded linear operators and holomorphic operator-valued functions. We focus on bounds for perturbed eigenvalues, eigenvectors and invariant subspaces and provide simplified proofs of some well known results. We present a comprehensive perturbation analysis of invariant subspaces of matrices. For bounded linear operators we discuss, among other things, the effect of analytic perturbation on the discrete eigenvalues and spectral projections. We also briefly discuss analytic spectral perturbation theory for holomorphic operator-valued functions.