{
  "abstract": "In 1997 Guo Wuwen proved that if $(λ_1,λ_2,\\dots,λ_n)$ is the spectrum of a nonnegative matrix with Perron eigenvalue $λ_1$ and $λ_2$ real, then $(λ_1+t,λ_2\\pm t,λ_3,\\dots,λ_n)$ is the spectrum of a nonnegative matrix for every $t\\ge 0$. Whether the same holds for symmetric nonnegative matrices has remained an open question. We give an affirmative answer, in a slightly stronger form: if $A$ is a symmetric nonnegative matrix with eigenvalues $λ_1\\ge\\dots\\geλ_n$ and $d_1\\ge|d_2|+\\dots+|d_n|$, then there is a symmetric matrix $B\\ge A$ with eigenvalues $λ_1+d_1,\\dots,λ_n+d_n$, and $B$ has the same diagonal as $A$ when $d_1+\\dots+d_n=0$. This leads us to call a symmetrically realizable list Guo-extreme if none of Guo's moves can be reversed within the set of symmetrically realizable lists. Every symmetrically realizable list is a Guo-extreme list plus a vector from the cone of Guo's moves. Our second main result is a certificate, modelled on a theorem of Laffey on extreme symmetric matrices: if $A$ is irreducible and every nonnegative symmetric matrix that commutes with $A$ and vanishes wherever $A$ is positive is a multiple of the identity, then the spectrum of $A$ can be moved by a small amount in every trace-preserving direction (in every direction, if $A$ has a positive diagonal entry) and remain symmetrically realizable; in particular it is not Guo-extreme.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14759",
  "author_total": 1,
  "authors": [
    {
      "name": "Helena Šmigoc"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7krvyqxn2m0hz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14759",
  "fulltext_url": null,
  "id": "7krvyqxn2m0hz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14759",
  "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",
      "math.SP"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7221754855",
  "pdf_url": "https://arxiv.org/pdf/2610.14759",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7krvyqxn2m0hz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14759",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14759"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A note on Guo's perturbation result for symmetric nonnegative matrices",
  "title_text": "A note on Guo's perturbation result for symmetric nonnegative matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7krvyqxn2m0hz",
  "venue": "arXiv",
  "versions": []
}