{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29009",
  "author_total": 2,
  "authors": [
    {
      "name": "Lei-Hong Zhang"
    },
    {
      "name": "Ren-Cang Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bv3gwnzz7xxra.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29009",
  "fulltext_url": null,
  "id": "bv3gwnzz7xxra",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29009",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7204917820",
  "pdf_url": "https://arxiv.org/pdf/2608.29009",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bv3gwnzz7xxra.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.29009",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29009"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix",
  "title_text": "A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/bv3gwnzz7xxra",
  "venue": "arXiv",
  "versions": []
}