Matrix Perturbation Theory in the Tangent Space of Isospectral Matrices

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.