{
  "id": "2406.11113",
  "version": "v1",
  "published": "2024-06-17T00:22:00.000Z",
  "updated": "2024-06-17T00:22:00.000Z",
  "title": "Matrix periods and competition periods of Boolean Toeplitz matrices II",
  "authors": [
    "Gi-Sang Cheon",
    "Bumtle Kang",
    "Suh-Ryung Kim",
    "Homoon Ryu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "This paper is a follow-up to the paper [Matrix periods and competition periods of Boolean Toeplitz matrices, {\\it Linear Algebra Appl.} 672:228--250, (2023)]. Given subsets $S$ and $T$ of $\\{1,\\ldots,n-1\\}$, an $n\\times n$ Toeplitz matrix $A=T_n\\langle S ; T \\rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \\in S$ or $i-j \\in T$. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices $A=T_n\\langle S; T \\rangle$ satisfying the condition ($\\star$) $\\max S+\\min T \\le n$ and $\\min S+\\max T \\le n$ are $d^+/d$ and $1$, respectively, where $d^+= \\gcd (s+t \\mid s \\in S, t \\in T)$ and $d = \\gcd(d, \\min S)$. In this paper, we claim that even if ($\\star$) is relaxed to the existence of elements $s \\in S$ and $t \\in T$ satisfying $s+t \\le n$ and $\\gcd(s,t)=1$, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy ($\\star$) but the relaxed condition. For example, for any positive integers $k, n$ with $2k+1 \\le n$, it is easy to see that $T_n\\langle k, n-k;k+1, n-k-1 \\rangle$ does not satisfies ($\\star$) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence $\\{A^m(A^T)^m\\}_{m=1}^\\infty$ is $T_n\\langle d^+,2d^+, \\ldots, \\lfloor n/d^+\\rfloor d^+\\rangle$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-17T00:22:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "toeplitz matrix",
      "boolean toeplitz matrices",
      "matrix period",
      "competition period",
      "linear algebra appl"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}