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arXiv · 2508.08203

A Note on Eigenvalues of Perturbed Hermitian Matrices

Chi-Kwong Li·Ren-Cang Li
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Abstract

Let $$ A=\left(\begin{array}{cc} H_1 & E^*\\ E & H_2\end{array}\right) \quad \hbox{ and } \quad \wtd A=\left(\begin{array}{cc} H_1 & O\\ O & H_2\end{array}\right)$$ be two $N$-by-$N$ Hermitian matrices with eigenvalues $λ_1 \ge \cdots \ge λ_{N}$ and $\wtd λ_1 \ge \cdots \ge \wtd λ_N$, respectively. \iffalse There are two kinds of perturbation bounds on $|λ_i - \wtd λ_i|$: $|λ_i- \wtd λ_i| \le \|E\|$, where $\|E\|$ is the largest singular value of $\|E\|$, regardless of $H_i$'s spectral distributions, and $|λ_i - \wtd λ_i| \le \|E\|^2/η$, where $η$ is the minimum gap between $H_i$'s spectra. \end{enumerate} Bounds of the first kind overestimate the changes when $\|E\|\llη$ while those of the second kind may blow up when $η$ is too tiny. \fi Denote by $\|E\|$ the spectral norm of the matrix $E$, and $η$ the spectral gap between the spectra of $H_1$ and $H_2$. It is shown that $$ |λ_i - \wtd λ_i| \le {2\|E\|^2 \over η+\sqrt{η^2+4\|E\|^2}} \, , $$ which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations.

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Chi-Kwong Li, Ren-Cang Li. 2025-08-11. A Note on Eigenvalues of Perturbed Hermitian Matrices. https://doi.org/10.1016/j.laa.2004.08.026

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