A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix

Abstract

Let $$A=\left[\begin{array}{cc}{H}_1& E^{\ast }\\ E& H_2\end{array}\right]\text{and}\tilde{A}=\left[\begin{array}{cc}{H}_1& 0\\ 0& H_2\end{array}\right]$$ be two partitioned Hermitian matrices, where $\tilde{A}$ is obtained from $A$ by simply dropping the off-diagonal blocks, and let $\eta$ be the gap between the spectra $\mathrm{eig}(H_1)$ of $H_1$ and $\mathrm{eig}(H_2)$ of $H_2$. Define, for $\delta \ge 0$ and $\epsilon \ge 0$, $$ ϕ(δ,ε)= \begin{cases} 2ε/(δ+\sqrt{δ^2+4ε^2}), &\quad\mbox{if $(δ,ε)\ne (0,0)$}, 1, &\quad\mbox{if $(δ,ε) = (0,0)$}, \end{cases} $$ and let $V=A-\tilde{A}$ and ${\epsilon }_2=\Vert E{\Vert }_2=\Vert V{\Vert }_2$, the matrix spectral norm. Li and Li [{\em Linear Algebra Appl.}, 395:183--190, 2005] established a sharp spectral-norm bound on the changes in the eigenvalues of $A$: $$\Vert \mathrm{diag}(\lambda (A)-\lambda (\tilde{A})){\Vert }_2\le \varphi (\eta ,{\epsilon }_2)\Vert E{\Vert }_2,$$ where $\lambda (A)$ is the vector whose components are the eigenvalues of $A$ in descending order and similarly for $\lambda (\tilde{A})$. The goal of this paper is to resolve the question: how far an extension of this result in the form $$\Vert \mathrm{diag}(\lambda (A)-\lambda (\tilde{A})){\Vert }_{\mathrm{UI}}\le \varphi (\eta ,{\epsilon }_2)\Vert A-\tilde{A}{\Vert }_{\mathrm{UI}}$$ remains valid for some or all unitarily invariant norms $\Vert \cdot {\Vert }_{\mathrm{UI}}$? Two results are obtained: (a) the extension holds for any $Q$-norm, a subclass of unitarily invariant norms that encompasses the Schatten $p$-norm for $2\le p\le \infty$ (particularly, the Frobenius norm and the spectral norm included), and (b) the extension holds for any unitarily invariant norm if $\mathrm{rank}(E)\le 1$. It is demonstrated that the equality is attained on the $2\times 2$ matrix $A$.

17 pages