{
  "abstract": "The $S$-matrix conjecture was formulated by Sloane and Harwit in 1976, motivated by an $A$-optimal design problem arising in spectroscopy. It states that if $A$ is a nonsingular real $n\\times n$ matrix whose entries lie in $[0,1]$, then $$\\left\\|A^{-1}\\right\\|_F\\ge\\frac{2n}{n+1}.$$ Moreover, equality holds if and only if $A$ is an $S$-matrix. In this paper, we prove the conjecture by combining a variational trace inequality in odd dimensions with a sharp range-constrained estimate for the Moore--Penrose inverse of a centered matrix in even dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20306",
  "author_total": 1,
  "authors": [
    {
      "name": "Teng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fxjbejx5cm17t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20306",
  "fulltext_url": null,
  "id": "fxjbejx5cm17t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20306",
  "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"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7213601278",
  "pdf_url": "https://arxiv.org/pdf/2609.20306",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fxjbejx5cm17t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20306",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20306"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The $A$-optimal design problem and the $S$-matrix conjecture",
  "title_text": "The A-optimal design problem and the S-matrix conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/fxjbejx5cm17t",
  "venue": "arXiv",
  "versions": []
}