A note on the sequence of consecutive powers of a nonnegative matrix in max algebra (Q5940032)

A max algebra consists of a set of nonnegative numbers with sum \(a\oplus b=\max\{a,b\}\) and the standard product \(ab\) for \(a,b\geq 0\). The product of matrices \(A=(a_{st})\) and \(B=(b_{st})\) is denoted by \(A\otimes B\) where \((A\otimes B)_{st}=\max_k\{a_{sk}b_{kt}\}\). A sequence \(\{A_i\}\) of real \(n\times n\) matrices is called asymptotically \(p\)-periodic if \(\lim_{k\to\infty}A_{j+kp}=\tilde A_j\) exists for \(1\leq j\leq p\). The minimal such \(p\) is called the asymptotic period of the sequence. For any nonnegative \(n\times n\) matrix \(A=(a_{st})\) with \(a_{st}\leq 1\), there exists a Boolean matrix \(\overline A\) with NEWLINE\[NEWLINE(\overline A)_{st}=\begin{cases} 1&\text{if \(A_{st}=1\)}\\ 0&\text{otherwise}\end{cases}NEWLINE\]NEWLINE By using the property of Boolean matrices, the authors prove that the sequence of consecutive powers \(\{A^n_{\otimes}=A\otimes\dots\otimes A\}\) has asymptotic period \(p\) if and only if the Boolean matrix \(\overline A\) has period \(p\).