The Jordan canonical form for a class of zero-one matrices (Q636258)

Given a function \(f:{\mathbb N} \to {\mathbb N}\) and an \(n \times n\) matrix \(A_n = (a_{ij} )\) defined by \(a_{ij} = 1\) if \(i = f(j)\) for some \(i\) and \(j\) and \(a_{ij} = 0\) otherwise, in this paper it is shown that the Jordan block structure of \(A_n\) can be described in terms of the cycles and chains of the directed graph for which \(A_n\) is the adjacency matrix. Finally, this result is illustrated by several examples including a connection with the Collatz \(3n + 1\) conjecture.