A Note on Eigenvalues of Perturbed Hermitian Matrices

Let $$A=\left(\begin{array}{cc}{H}_1& E^{\ast }\\ E& H_2\end{array}\right)\text{ and }\text{wtd}A=\left(\begin{array}{cc}{H}_1& O\\ O& H_2\end{array}\right)$$ be two $N$-by-$N$ Hermitian matrices with eigenvalues ${\lambda }_1\ge \cdots \ge {\lambda }_N$ and $\text{wtd}{\lambda }_1\ge \cdots \ge \text{wtd}{\lambda }_N$, respectively. \iffalse There are two kinds of perturbation bounds on $\mathrm{\mid }{\lambda }_i-\text{wtd}{\lambda }_i\mathrm{\mid }$: $\mathrm{\mid }{\lambda }_i-\text{wtd}{\lambda }_i\mathrm{\mid }\le \mathrm{\parallel }E\mathrm{\parallel }$, where $\mathrm{\parallel }E\mathrm{\parallel }$ is the largest singular value of $\mathrm{\parallel }E\mathrm{\parallel }$, regardless of $H_i$’s spectral distributions, and $\mathrm{\mid }{\lambda }_i-\text{wtd}{\lambda }_i\mathrm{\mid }\le \mathrm{\parallel }E{\mathrm{\parallel }}^2\mathrm{/}\eta$, where $\eta$ is the minimum gap between $H_i$’s spectra. \end{enumerate} Bounds of the first kind overestimate the changes when $\mathrm{\parallel }E\mathrm{\parallel }\ll \eta$ while those of the second kind may blow up when $\eta$ is too tiny. \fi Denote by $\mathrm{\parallel }E\mathrm{\parallel }$ 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 $$\mathrm{\mid }{\lambda }_i-\text{wtd}{\lambda }_i\mathrm{\mid }\le \frac{2\mathrm{\parallel }E{\mathrm{\parallel }}^2}{\eta +\sqrt{{\eta }^2+4\mathrm{\parallel }E{\mathrm{\parallel }}^2}},$$ which improves all the existing results. Similar bounds are obtained for singular values of matrices under block perturbations.