Matrix periods and competition periods of Boolean Toeplitz matrices II

TL;DR

This paper extends previous results on the periods of Boolean Toeplitz matrices by relaxing conditions and analyzing the limit behavior of matrix sequences, revealing new structural insights.

Contribution

It generalizes earlier findings by showing the same period results hold under a relaxed condition, and characterizes the limit of specific matrix sequences.

Findings

Matrix period is d^+/d under relaxed conditions.

Competition period remains 1 for the matrices considered.

Limit of the matrix sequence is a Toeplitz matrix with specific entries.

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 $T_n\langle d^+,2d^+, \ldots, \lfloor n/d^+\rfloor d^+\rangle$.