{
  "abstract": "We classify the simultaneous half-bandwidths $p,q$ of a symmetric oscillatory matrix $A$ and its inverse. At order $n\\ge2$, the possible pairs are $(1,n-1)$, $(n-1,1)$, and all pairs with $2\\le p,q\\le n-1$. Let $e_k(A)$ be the least positive integer $m$ for which all minors of order $k$ of $A^m$ are positive, and let $e(A)=\\max_{1\\le k<n}e_k(A)$. Put $a=\\min\\{p,q\\}$ and $b=\\max\\{p,q\\}$. At every feasible pair of actual half-bandwidths, \\[ \\max_A e(A)= \\max\\left\\{\\left\\lfloor\\frac n2\\right\\rfloor, n-a-\\left\\lfloor\\frac{\\max\\{n-p-q,0\\}}{b}\\right\\rfloor\\right\\}. \\] We also determine the maximum of $\\sum_{k=1}^{n-1}e_k(A)$ for every pair. When $p+q\\ge n$, one realization maximizes all $e_k$ simultaneously. Write $A=U^TΔU$, with $U$ unit upper triangular and totally nonnegative, and $Δ$ positive diagonal. Let $\\ell(U)$ be the least number of elementary upper bidiagonal factors with positive parameters whose product is $U$. At fixed $(p,q)$, its minimum is $\\max\\{n-1,p+q-1\\}$. One symmetric integer matrix of determinant one simultaneously attains this minimum, the maximum exponent, and the maximum exponent sum. We prove $\\ell(U^{r+1})-\\ell(U^r)=\\#\\{k:1\\le k<n,\\ e_k(A)>r\\}$ for every integer $r\\ge\\lfloor(n-1)/2\\rfloor$, with the starting point optimal. With $δ=\\ell(U)-n+1$, the exponent sum satisfies $δ\\le n(n-1)/2-\\sum_{k=1}^{n-1}e_k(A) \\le\\lfloor(n-1)/2\\rfloorδ$, with both constants sharp. The lower deficit bound is attained at every admissible factor count. Within the symmetric class, a total sum of $n(n-1)/2$ characterizes basic matrices. In the basic oscillatory class, without a symmetry assumption, a positive integer vector $(f_1,\\ldots,f_{n-1})$ is realizable if and only if $f_k\\le\\max\\{k,n-k\\}$ and $f_k+f_{n-k}\\ge n$ for $1\\le k<n$, and $\\lvert f_{k+1}-f_k\\rvert\\le1$ for $1\\le k<n-1$. Explicit factor orders realize every such vector.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35846",
  "author_total": 1,
  "authors": [
    {
      "name": "Wei Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6nmekx03pvkjk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35846",
  "fulltext_url": null,
  "id": "6nmekx03pvkjk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35846",
  "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": "W7214937646",
  "pdf_url": "https://arxiv.org/pdf/2609.35846",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6nmekx03pvkjk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.35846",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35846"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Exponent Profiles of Oscillatory Matrices under Matrix and Inverse Sparsity",
  "title_text": "Exponent Profiles of Oscillatory Matrices under Matrix and Inverse Sparsity",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:26Z",
  "url": "https://research.draconitic.com/paper/6nmekx03pvkjk",
  "venue": "arXiv",
  "versions": []
}