{
  "abstract": "We present a new proof of a classical theorem of Dieudonné: if a linear space of $n\\times n$ matrices consists entirely of singular matrices, then its dimension is at most $n^2-n$. Our proof is based on a surprising ``duality'' argument: we prove this universal upper bound by exhibiting a single matrix space that serves as a lower bound for a related problem. Interestingly, this approach only works for certain fields, but we use model-theoretic arguments to obtain the same result for all fields. We hope that this approach can be generalized to provide new duality-based proofs of other classical theorems on matrix spaces, and give some preliminary results in this direction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29177",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuval Wigderson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1wqkbhwg2gvy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29177",
  "fulltext_url": null,
  "id": "c1wqkbhwg2gvy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29177",
  "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": "W7214360817",
  "pdf_url": "https://arxiv.org/pdf/2609.29177",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1wqkbhwg2gvy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29177",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29177"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Duality for matrix space questions",
  "title_text": "Duality for matrix space questions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/c1wqkbhwg2gvy",
  "venue": "arXiv",
  "versions": []
}