{
  "abstract": "In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate $$ \\min_{W\\ \\mathrm{nonsingular}}\\|W\\circ G\\|_\\mathrm{ui}\\,\\|W^{-1}\\circ G^\\mathrm{T}\\|_\\mathrm{ui} \\quad\\mbox{and}\\quad \\min_{P\\ \\mathrm{permutation}}\\|P\\circ G\\|_\\mathrm{ui}^2, $$ where $\\|\\cdot\\|_\\mathrm{ui}$ denotes a unitarily invariant norm and $\\circ$ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every $G$, then $\\|\\cdot\\|_\\mathrm{ui}$ must be uniformly equivalent to the Frobenius norm $\\|\\cdot\\|_\\mathrm{F}$. We prove that the two quantities coincide when $\\|\\cdot\\|_\\mathrm{ui}=\\|\\cdot\\|_\\mathrm{F}$: for every complex matrix $G$, $$ \\min_{W\\ \\mathrm{nonsingular}} \\|W\\circ G\\|_\\mathrm{F}\\,\\|W^{-1}\\circ G^\\mathrm{T}\\|_\\mathrm{F} =\\min_{P\\ \\mathrm{permutation}}\\|P\\circ G\\|_\\mathrm{F}^2. $$ The associated conjecture therefore holds with the optimal constant $c=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25552",
  "author_total": 2,
  "authors": [
    {
      "name": "Lei-Hong Zhang"
    },
    {
      "name": "Ren-Cang Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84a9gbdyx7z7h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25552",
  "fulltext_url": null,
  "id": "84a9gbdyx7z7h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25552",
  "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": "W7214082215",
  "pdf_url": "https://arxiv.org/pdf/2609.25552",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84a9gbdyx7z7h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25552",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25552"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "On a Conjecture Related to Eigenvalue Perturbations",
  "title_text": "On a Conjecture Related to Eigenvalue Perturbations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/84a9gbdyx7z7h",
  "venue": "arXiv",
  "versions": []
}