{
  "abstract": "The quantity $q(A\\circ A^{-1})$, the minimum modulus of the eigenvalues of $A\\circ A^{-1}$, arises naturally in connection with positive diagonal symmetrizability. For an invertible $\\mathbf{M}$-matrix $A$ of order $n$, the classical bounds $\\frac{2}{n}\\leq q(A\\circ A^{-1})\\leq 1$ are known. We discuss the sharpness of the lower bound $\\frac{2}{n}$ and investigate the converse of a related result involving the Jacobi iteration matrix. In particular, we show that $ρ(J_{A_k})\\to 1$ does not, in general, imply $q(A_k\\circ A_k^{-1})\\to \\frac{2}{n}$, and identify a class for which this implication holds. We then turn to invertible $\\mathbf{H}$-matrices, a broader class that contains invertible $\\mathbf{M}$-matrices. We show that $A\\circ A^{-1}$ is an invertible $\\mathbf{H}$-matrix whenever $A$ is an invertible $\\mathbf{H}$-matrix. In contrast to the $\\mathbf{M}$-matrix setting, $q(A\\circ A^{-1})$ can be arbitrarily close to zero. However, replacing $A^{-1}$ by the inverse of the comparison matrix restores the classical lower bound: we prove that $q(A\\circ\\mathcal{M}(A)^{-1})\\geq \\frac{2}{n}$ and obtain further bounds involving the Jacobi iteration matrix of $\\mathcal{M}(A)$. Finally, for positive diagonally symmetrizable invertible $\\mathbf{H}$-matrices, we establish the upper bound $q(A\\circ A^{-1})\\leq1$ and, in the irreducible case, characterize when equality occurs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24019",
  "author_total": 3,
  "authors": [
    {
      "name": "Bharat Pratap Chauhan"
    },
    {
      "name": "Samir Mondal"
    },
    {
      "name": "Sushmitha P"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c16c7mmmsf0tr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24019",
  "fulltext_url": null,
  "id": "c16c7mmmsf0tr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24019",
  "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.FA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7214031691",
  "pdf_url": "https://arxiv.org/pdf/2609.24019",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c16c7mmmsf0tr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24019",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24019"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Bounds on the Minimum Eigenvalue Modulus for Hadamard Products of $\\mathbf{M}$- and $\\mathbf{H}$-Matrices and Their Inverses",
  "title_text": "Bounds on the Minimum Eigenvalue Modulus for Hadamard Products of M- and H-Matrices and Their Inverses",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/c16c7mmmsf0tr",
  "venue": "arXiv",
  "versions": []
}