A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix
Year: 2026
Type: preprint
Abstract: Let $$ A=\begin{bmatrix} H_1 & E^* \\ E & H_2 \end{bmatrix} \quad\text{and}\quad \widetilde A=\begin{bmatrix} H_1 & 0 \\ 0 & H_2 \end{bmatrix} $$ be partitioned Hermitian matrices, with $\widetilde A$ obtained by dropping the off-diagonal blocks of $A$. Let $η$ be the minimum distance between the spectra of $H_1$ and $H_2$; these spectra may interlace. Define, for $δ,ε\ge0$, $$ ϕ(δ,ε)= \begin{cases} 2ε/(δ+\sqrt{δ^2+4ε^2}), &\quad\mbox{if $(δ,ε)\ne (0,0)$}, 1, &\quad\mbox{if $(δ,ε) = (0,0)$}, \end{cases} $$ and let $\widetilde E=A-\widetilde A$ and $ε_2=\|E\|_2=\|\widetilde E\|_2$, the matrix spectral norm. Li and Li [\emph{Linear Algebra Appl.}, 395:183--190, 2005] established a sharp spectral-norm bound on the changes in the eigenvalues of $A$: $$ \big\|\mathrm{diag}\big(\pmbλ(A)-\pmbλ(\widetilde A)\big)\big\|_2 \le ϕ(η,ε_2)\,\|E\|_2, $$ where $\pmbλ(A)$ is the vector whose components are the eigenvalues of $A$ in descending order and similarly for $\pmbλ(\widetilde A)$. In this paper, this bound is extended to every unitarily invariant norm $\|\cdot\|_\mathrm{ui}$: $$ \big\|\mathrm{diag}\big(\pmbλ(A)-\pmbλ(\widetilde A)\big)\big\|_\mathrm{ui} \le ϕ(η,ε_2)\,\|A-\widetilde A\|_\mathrm{ui}. $$ A $2\times2$ example attains equality for every such norm. An analogous bound for singular values is also established when the off-diagonal blocks of a rectangular matrix are dropped.
Source: arXiv (Cornell University)
Authors: Zhang, Lei-Hong, Li, Ren-Cang
Institutions: Soochow University, The University of Texas at Arlington
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204917820