{
  "abstract": "Boros and Rózsa (Linear Algebra Appl. 421 (2007) 407--416) reduced the singular value problem for the odd-order Sylvester--Kac matrix $P\\in\\mathbb{R}^{(2m+1)\\times(2m+1)}$ to those for $Q^{(m)}\\in\\mathbb{R}^{(m+1)\\times m}$ and $S^{(m)}\\in\\mathbb{R}^{m\\times(m+1)}$ via a permutation similarity transformation. They noted that $(Q^{(m)})^{T}Q^{(m)}-S^{(m)}(S^{(m)})^{T}$ is positive definite, and also conjectured that $σ_i(S^{(m)})\\geσ_{i+1}(Q^{(m)})$ for $1\\le i\\le m-1$ and $m\\ge2$. In this paper, we prove that these inequalities are strict and establish the strict interlacing relation \\begin{equation*} σ_1(Q^{(m)})>σ_1(S^{(m)})> \\cdots>σ_m(Q^{(m)})> σ_m(S^{(m)})>0 \\end{equation*} for all $m\\ge1$. Here, $σ_i(\\cdot)$ denotes the $i$-th largest singular value of a matrix.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21702",
  "author_total": 2,
  "authors": [
    {
      "name": "Fuying Tang"
    },
    {
      "name": "Shuo Shi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24ya1yaj8r658.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21702",
  "fulltext_url": null,
  "id": "24ya1yaj8r658",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21702",
  "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.FA"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.21702",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24ya1yaj8r658.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21702",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21702"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices",
  "title_text": "A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/24ya1yaj8r658",
  "venue": "arXiv",
  "versions": []
}