{
  "abstract": "Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\\mathcal{E}$, invariant by complex conjugation and containing real matrices, does the subset of $\\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius distance always contain a real matrix? We give negative answers when $\\mathcal{E}$ is either the set of normal matrices (solving a question posed by N. Higham) or the set of matrices whose eigenvalues lie in the closed left half-plane (solving a question posed by the first author and F. Poloni). For the latter problem, we also argue that the ratio between the real and complex distances is unbounded in every dimension $n\\geq 3$, and this holds for every distance induced by a unitarily invariant norm. For both problems and $n \\geq 3$, we show that a real input $A$ uniformly drawn from the unit sphere $\\|A\\|_F=1$ has no real minimizer over $\\mathcal{E}$ with probability strictly between $0$ and $1$. Beyond these two cases, we prove for an arbitrary nonempty closed target sets $\\mathcal{E}$ that, for almost every real input, either $\\mathcal{E}$ contains a unique nearest matrix which is real, or every nearest matrix in $\\mathcal{E}$ is nonreal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14828",
  "author_total": 2,
  "authors": [
    {
      "name": "Vanni Noferini"
    },
    {
      "name": "Matvei Zhukov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdrbd6gq1w3m6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14828",
  "fulltext_url": null,
  "id": "fdrbd6gq1w3m6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14828",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7213351601",
  "pdf_url": "https://arxiv.org/pdf/2609.14828",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdrbd6gq1w3m6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.14828",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14828"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Matrix nearness problems: Do real inputs admit real solutions?",
  "title_text": "Matrix nearness problems: Do real inputs admit real solutions?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/fdrbd6gq1w3m6",
  "venue": "arXiv",
  "versions": []
}