{
  "abstract": "Let $m,n\\ge 2$. For $m\\times n$ complex matrices $A$, $C$ and an $n\\times m$ matrix $B$, Nobori (Linear Algebra Appl. 725 (2025) 135--144) proposed the following conjecture on the generalized Böttcher-Wenzel inequality: $$ \\|ABC-CBA\\|_F^2 \\le 2\\|B\\|_2^2\\|A\\|_{(2),2}^2\\|C\\|_F^2, $$ where $\\|\\cdot\\|_F$ is the Frobenius norm, $\\|\\cdot\\|_2$ is the spectral norm, and $\\|\\cdot\\|_{(2),2}$ is the $(2,2)$-norm defined by $\\|X\\|_{(2),2}=\\sqrt{σ_1(X)^2+σ_2(X)^2}$, in which $σ_1(X)$ and $σ_2(X)$ are the largest and second largest singular values of $X$, respectively. This paper proves the conjecture and, as a consequence, establishes the Kronecker product inequality that Nobori derived from it. In addition, we show that for each fixed $m,~n\\ge 2$, the constant $2$ is sharp. We also give necessary and sufficient conditions for equality, namely, the equality condition in the Böttcher--Wenzel inequality and some matrix identities derived from a fixed singular value decomposition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23520",
  "author_total": 2,
  "authors": [
    {
      "name": "Shuo Shi"
    },
    {
      "name": "Juan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2b047q7g6sytj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23520",
  "fulltext_url": null,
  "id": "2b047q7g6sytj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23520",
  "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.RA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7214040932",
  "pdf_url": "https://arxiv.org/pdf/2609.23520",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2b047q7g6sytj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23520",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23520"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture",
  "title_text": "Proof of Nobori's generalized Böttcher-Wenzel inequality conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/2b047q7g6sytj",
  "venue": "arXiv",
  "versions": []
}