A Sharp Version of Kahan''s Theorem on Clustered Eigenvalues

Let $n\times n$ Hermitian matrix $A$ have eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_n$, and let $k\times k$ Hermitian matrix $H$ have eigenvalues $\mu_1\le\mu_2\le\cdots\le\mu_k$, and let $Q$ be an $n\times k$ matrix having full column rank, so $1\le k\le n$. It is proved that there exist $k$ eigenvalues $\lambda_{i_1}\le\lambda_{i_2}\le\cdots\le\lambda_{i_k}$ of $A$ such that $$ \max_{1\le j\le k}|\mu_j-\lambda_{i_j}| \le\frac c{\sigma_{\min}(Q)}\|AQ-QH\|_2, $$ always holds with $c=1$, where $\sigma_{\min}(Q)$ is the smallest singular value of $Q$ and $\|\cdot\|_2$ denotes the biggest singular value of a matrix. The inequality was proved for $c\le\sqrt 2$ in 1967 by Kahan who also conjectured that it should be true for $c=1$.