Matrix periods and competition periods of Boolean Toeplitz matrices II

This paper is a follow-up to the paper [Matrix periods and competition periods of Boolean Toeplitz matrices, { Linear Algebra Appl.} 672:228--250, (2023)]. Given subsets S and T of \1,…,n-1\, an n× n Toeplitz matrix A=Tₙ S ; T is defined to have $1$ as the $(i,j)$-entry if and only if j-i ∈ S or i-j ∈ T. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices A=Tₙ S; T satisfying the condition () max S+min T ≤ n and min S+max T ≤ n are d⁺/d and $1$, respectively, where d⁺= (s+t s ∈ S, t ∈ T) and d = (d, min S). In this paper, we claim that even if () is relaxed to the existence of elements s ∈ S and t ∈ T satisfying s+t ≤ n and (s,t)=1, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy () but the relaxed condition. For example, for any positive integers $k, n$ with 2k+1 ≤ n, it is easy to see that Tₙ k, n-k;k+1, n-k-1 does not satisfies () but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence ᵐ(AT)ᵐ\ₘ₌₁^∞ is Tₙ d⁺,2d⁺, …, n/d⁺ d⁺.