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

Abstract

Let A= H₁ & E^* E & H₂ and Ã= H₁ & 0 0 & H₂ be partitioned Hermitian matrices, with Ã obtained by dropping the off-diagonal blocks of A. Let η be the minimum distance between the spectra of H₁ and H₂; these spectra may interlace. Define, for δ,ε≥0, ϕ(δ,ε)= 2ε/(δ+√(δ²+4ε²)), & if $(δ,ε)≠ (0,0)$, 1, & if $(δ,ε) = (0,0)$, and let Ẽ=A−Ã and ε₂=|E|₂=|Ẽ|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: |diag(pmbλ(A)−pmbλ(Ã))|₂ ≤ ϕ(η,ε₂) |E|₂, where pmbλ(A) is the vector whose components are the eigenvalues of A in descending order and similarly for pmbλ(Ã). In this paper, this bound is extended to every unitarily invariant norm |·|ui: |diag(pmbλ(A)−pmbλ(Ã))|ui ≤ ϕ(η,ε₂) |A−Ã|ui. A 2×2 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.

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BibTeX RIS

BibTeX

@misc{zhang2026sharp,
  title = {{A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix}},
  author = {Lei-Hong Zhang and Ren-Cang Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.29009},
  eprint = {2608.29009},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29009},
}

RIS

TY  - UNPB
TI  - A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix
AU  - Lei-Hong Zhang
AU  - Ren-Cang Li
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.29009
UR  - https://arxiv.org/abs/2608.29009
AB  - Let A= H_1 & E^* E & H_2 and Ã= H_1 & 0 0 & H_2 be partitioned Hermitian matrices, with Ã 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≥0, ϕ(δ,ε)= 2ε/(δ+√(δ^2+4ε^2)), & if $(δ,ε)≠ (0,0)$, 1, & if $(δ,ε) = (0,0)$, and let Ẽ=A−Ã and ε_2=|E|_2=|Ẽ|_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: |diag(pmbλ(A)−pmbλ(Ã))|_2 ≤ 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(Ã). In this paper, this bound is extended to every unitarily invariant norm |·|_ui: |diag(pmbλ(A)−pmbλ(Ã))|_ui ≤ ϕ(η,ε_2) |A−Ã|_ui. A 2×2 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.
ER  -

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