{
  "abstract": "For a finite abelian group $G$, let $μ_2(G)$ be the minimum number of positive eigenvalues of an entrywise positive real symmetric group-circulant matrix whose entrywise square is positive definite. We prove that $μ_2(G)\\le5$ precisely when $|G|\\le15$, and that $μ_2(G)=6$ precisely when $|G|=16$ or $G$ is cyclic of order $18$, $20$ or $22$. For the noncyclic groups of order at most fifteen, the minimum is three for $C_2^2$ and five otherwise. If the positive Fourier support of such a matrix is $\\{0,\\pm r_1,\\ldots,\\pm r_d\\}$, where $d\\ge1$ and all $2d+1$ displayed elements are distinct, then there are integers $α_1,\\ldots,α_d$, not all zero, such that $\\sum_iα_i r_i=0$ and $\\sum_i|α_i|\\le\\max\\{5,d+2\\}$. The bound five is sharp for $d=2$ and $d=3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22744",
  "author_total": 1,
  "authors": [
    {
      "name": "Wei Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3t07wejyay8zh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22744",
  "fulltext_url": null,
  "id": "3t07wejyay8zh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22744",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7214001695",
  "pdf_url": "https://arxiv.org/pdf/2609.22744",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3t07wejyay8zh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2609.22744",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22744"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups",
  "title_text": "Positive Eigenvalues of Hadamard Square Roots on Finite Abelian Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/3t07wejyay8zh",
  "venue": "arXiv",
  "versions": []
}