Matrix Perturbation Theory in the Tangent Space of Isospectral Matrices

TL;DR

This paper extends eigenvalue and eigenvector perturbation theory to matrices with perturbations in the form of commutators, providing new bounds and insights for structured perturbations in various matrix settings.

Contribution

It generalizes existing results by analyzing perturbations of the form E = AB - BA, offering new eigenvalue and eigenvector bounds and extending to block-diagonal and Jordan block cases.

Findings

Eigenvalue perturbation order is  imes orm{E}^2 / ext{spectral gap}

Detailed analysis of matrix B's role in eigenvector perturbation

Extension to block-diagonal matrices with multiple eigenvalues and Jordan blocks

Abstract

Eigenvalue and eigenvector perturbation theory is a fundamental topic in several disciplines, including numerical linear algebra, quantum physics, and related fields. The central problem is to understand how the eigenvalues and eigenvectors of a matrix $A \in \mathbb{C}^{n \times n}$ change under the addition of a perturbation matrix $E \in \mathbb{C}^{n \times n}$. Much of the existing literature focuses on structured perturbations. For example, in [C.-K. Li and R.-C. Li, Linear Algebra Appl. 2005], the matrix $A$ is assumed to be Hermitian and block diagonal, while the perturbation $E$ is Hermitian and block off-diagonal. In this work, we investigate a different structured setting in which the perturbation has the commutator form $E = AB - BA$ for some matrix $B$, which we show to be a generalization of the block diagonal structure considered by Li and Li. First, we extend their main result by showing that the perturbation of the $i$-th eigenvalue of $A$, denoted by $\lambda_i$, is of order $\|E\|^2 / \eta_i$, where $\eta_i = \min_{j \neq i} |\lambda_i - \lambda_j|$ is the spectral gap associated with $\lambda_i$. Second, we provide a detailed analysis of the role played by the matrix $B$ in the perturbation of the eigenvectors. This analysis is further generalized to the case of block-diagonal matrices with multiple eigenvalues, as well as to perturbed singular values and eigenvalues of Jordan blocks.