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 $\hat{I}{»}_1\ge \cdots \ge \hat{I}{»}_N$ 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 $\Vert E\Vert$ is the largest singular value of $\Vert E\Vert$, regardless of $H_i$'s spectral distributions, and $|λ_i - \wtd λ_i| \le \|E\|^2/η$, where $\hat{I}\cdot$ is the minimum gap between $H_i$'s spectra. \end{enumerate} Bounds of the first kind overestimate the changes when $\Vert E\Vert \ll \hat{I}\cdot$ while those of the second kind may blow up when $\hat{I}\cdot$ is too tiny. \fi Denote by $\Vert E\Vert$ the spectral norm of the matrix $E$, and $\hat{I}\cdot$ 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.