{
  "abstract": "We classify linear maps $ϕ$ between possibly different rectangular matrix spaces over a field $\\mathbb F$ with $|\\mathbb F|\\ge5$ such that $\\operatorname{rank} ϕ(A)\\le2$ whenever $\\operatorname{rank} A\\le2$. Every such map has one of six forms. If its image contains a matrix of rank greater than two, then $ϕ(A)=MAN$ or $ϕ(A)=MA^{\\top}N$ for fixed matrices $M,N$. The remaining forms have images consisting entirely of matrices of rank at most two, including an exceptional form arising from $3\\times3$ alternating matrices. As a consequence, we characterize linear maps preserving rank exactly two. We also prove that rank-$k$ preservation implies rank-$k$ nonincrease when $|\\mathbb F|>k+2$, and that rank-$k$ nonincrease implies rank-$\\ell$ nonincrease for every admissible $\\ell\\ge k$ when $|\\mathbb F|>k+1$. The classification uses the fact that the Chow rank of the generic $3\\times3$ determinant is five, connecting the rank preserver problem with product decompositions of polynomials.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40995",
  "author_total": 2,
  "authors": [
    {
      "name": "Huajun Huang"
    },
    {
      "name": "Ming-Cheng Tsai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cz3z3dy70rvjz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40995",
  "fulltext_url": null,
  "id": "cz3z3dy70rvjz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40995",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40995",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cz3z3dy70rvjz.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40995",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40995"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Linear Rank Nonincreasing and Preserving Maps between Matrix Spaces",
  "title_text": "Linear Rank Nonincreasing and Preserving Maps between Matrix Spaces",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/cz3z3dy70rvjz",
  "venue": "arXiv",
  "versions": []
}