{
  "abstract": "For any integers $x$ and $y$, let $(x,y)$ and $[x,y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$, respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\\{x_1,...,x_n\\}$ be a set of $n$ distinct positive integers. Let $(f((x_i,x_j)))$ (abbreviated by $f(S)$) and $(f([x_i,x_j]))$ (abbreviated by $(f([S]))$) stand for the $n\\times n$ matrices whose $(i,j)-$entry is $(f((x_i,x_j)))$ and $(f([x_i,x_j]))$ respectively. In 1989, Beslin and Ligh gave a description of the lower triangular decomposition of $((x_i,x_j))$. In 1992, Bourque and Ligh showed that if $S$ is factor closed (i.e., S contains all positive divisors of any element of S), then the GCD matrix $((x_i,x_j))$ divides the LCM matrix $([x_i,x_j])$ (written as $((x_i,x_j))|([x_i,x_j])$) in the ring $M_n(\\mathbb{Z})$ of $n\\times n$ matrices over the integers. In this paper, we will show the diagonalization of $((x_i,x_j))$ and its applications. Our main new contributions are Theorems 4.1 and 4.2, which extend previous results to gcd-closed sets satisfying condition $\\mathcal{G}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07352",
  "author_total": 2,
  "authors": [
    {
      "name": "Wenzhong Lei"
    },
    {
      "name": "Han Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/62984dy9p74hy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07352",
  "fulltext_url": null,
  "id": "62984dy9p74hy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07352",
  "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.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212038962",
  "pdf_url": "https://arxiv.org/pdf/2609.07352",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/62984dy9p74hy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07352",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07352"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Normal Form of Smith's Matrices",
  "title_text": "The Normal Form of Smith's Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/62984dy9p74hy",
  "venue": "arXiv",
  "versions": []
}