{
  "abstract": "Complex matrix multiplication is typically computed using 4 real matrix multiplications (GEMMs) of the same size. The well-known 3M multiplication algorithm reduces this cost to 3 real GEMMs, together with quadratic time pre- and post-processing steps. In this paper, we reduce 3M to 2M for matrices with integer real and imaginary parts, performing complex GEMM with only 2 real GEMMs of the same size, along with quadratic time pre- and post-processing. For floating-point matrices, 2M multiplication combines naturally with the Ozaki-II scheme, yielding a practical, high-performance algorithm for computing a complex floating-point GEMM in roughly twice the time of a real GEMM of the same size. As corollaries, we derive new algorithms for symmetric rank-$k$ updates (SYRK/HERK) that internally use full rectangular GEMMs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05419",
  "author_total": 1,
  "authors": [
    {
      "name": "Peter Caday"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1g6ezfke7yqqe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05419",
  "fulltext_url": null,
  "id": "1g6ezfke7yqqe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05419",
  "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",
      "cs.DC"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.05419",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1g6ezfke7yqqe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.05419",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05419"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "The 2M Multiplication Algorithm for Complex Matrices",
  "title_text": "The 2M Multiplication Algorithm for Complex Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/1g6ezfke7yqqe",
  "venue": "arXiv",
  "versions": []
}