{
  "abstract": "Products of positive semidefinite matrices arise when successive steps of a randomized optimization method use different data points. Their average can have a negative eigenvalue even though every individual matrix is positive semidefinite. Lai and Lim proved at ICML 2020 that $\\sum_{i\\ne j}A_iA_j\\succeq-\\frac{1}{4}n(n-1)I$ whenever $A_i\\succeq0$ and $\\sum_i A_i\\preceq nI$, and conjectured that the constant is sharp for every $n\\ge2$. We give an explicit family of $n$ rank-one matrices in dimension $n$ that attains this bound for every $n$, proving that sharpness statement. The construction compresses one direction of an orthonormal basis by a factor $1/\\sqrt{2}$. We also give a short, self-contained proof of the known inequality and classify all equality cases: every extremizer contains a common invariant $n$-dimensional block equivalent to the construction. Consequently, dimension $n$ is necessary for equality, and positive definite matrices approach but never attain the lower endpoint. Finally, strongly convex quadratic objectives realize the extremal negative eigenvalue as a two-step expected gradient-descent update.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00155",
  "author_total": 1,
  "authors": [
    {
      "name": "Jinze Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/53pmfnsf7h9cg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00155",
  "fulltext_url": null,
  "id": "53pmfnsf7h9cg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00155",
  "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.OC"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7215947907",
  "pdf_url": "https://arxiv.org/pdf/2610.00155",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/53pmfnsf7h9cg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00155",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00155"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Sharpness and Rigidity of the Two-Factor Recht-Re Lower Bound",
  "title_text": "Sharpness and Rigidity of the Two-Factor Recht-Re Lower Bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/53pmfnsf7h9cg",
  "venue": "arXiv",
  "versions": []
}