{
  "abstract": "We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold $U(n)$ of unitary matrices of size n or on the flag manifold $U (n)/U (1)^n$. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28545",
  "author_total": 2,
  "authors": [
    {
      "name": "Vanni Noferini"
    },
    {
      "name": "Matvei Zhukov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2cx4j1p9s4ne2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28545",
  "fulltext_url": null,
  "id": "2cx4j1p9s4ne2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28545",
  "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"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7204834051",
  "pdf_url": "https://arxiv.org/pdf/2608.28545",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2cx4j1p9s4ne2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28545",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28545"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Closest Normal Matrix Found Again Using Riemannian Optimization",
  "title_text": "Closest Normal Matrix Found Again Using Riemannian Optimization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/2cx4j1p9s4ne2",
  "venue": "arXiv",
  "versions": []
}