A Note on Eigenvalues of Perturbed Hermitian Matrices

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 $\lambda_1 \ge \cdots \ge \lambda_{N}$ and $\wtd \lambda_1 \ge \cdots \ge \wtd \lambda_N$, respectively. \iffalse There are two kinds of perturbation bounds on $|\lambda_i - \wtd \lambda_i|$: $|\lambda_i- \wtd \lambda_i| \le \|E\|$, where $\|E\|$ is the largest singular value of $\|E\|$, regardless of $H_i$'s spectral distributions, and $|\lambda_i - \wtd \lambda_i| \le \|E\|^2/\eta$, where $\eta$ is the minimum gap between $H_i$'s spectra. \end{enumerate} Bounds of the first kind overestimate the changes when $\|E\|\ll\eta$ while those of the second kind may blow up when $\eta$ is too tiny. \fi Denote by $\|E\|$ the spectral norm of the matrix $E$, and $\eta$ the spectral gap between the spectra of $H_1$ and $H_2$. It is shown that $$ |\lambda_i - \wtd \lambda_i| \le {2\|E\|^2 \over \eta+\sqrt{\eta^2+4\|E\|^2}} \, , $$ which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations.