Eventual periodicity of the Smith forms of integer matrix powers

Abstract

We prove that the Smith forms of the powers of an integer square matrix behave in an eventually periodic manner. More precisely, if $\mathrm{SF}(M)$ denotes the Smith form of $M \in \mathbb{Z}^{m \times m}$, then for every $A \in \mathbb{Z}^{m \times m}$ there exist $n_0 \in \mathbb{N}$, an integer $T \geq 1$, and a constant diagonal matrix $D \in \mathbb{Z}^{m \times m}$ such that $n \geq n_0$ implies $\mathrm{SF}(A^{n+T})=D \cdot \mathrm{SF}(A^n)$. This provides an eventually affirmative answer to a conjecture posed in 2013 by R. Bruner. We also show that both $n_0$ and $T$ can be arbitrarily large.