So, say I have a matrix $A\in M_d(\mathbb{Z})$ and would like to describe the group $\lim(\mathbb{Z}^d,A)$, i.e. the limit of the stationary system
$$ \mathbb{Z}^d\to^A \mathbb{Z}^d \to^A \mathbb{Z}^d \dots $$
Furthermore, I know that $A$ is diagonizable, so I have matrices $P\in M_d(\mathbb{Z})$ and diagonal $D\in M_n(\mathbb{Z})$ such that $A=PDP^{-1}$, the entries of $D$ simply being the eigenvalues of $A$. Does this in any way help me get to grips with the group $\lim(\mathbb{Z}^d,A)$?
Now, if I'm lucky and $P^{-1}\in M_d(\mathbb{Z})$, I'm done, since the maps $P$ and $P^{-1}$ intertwine the diagrams
$$ \mathbb{Z}^d\to^A \mathbb{Z}^d \to^A \mathbb{Z}^d \dots $$
and
$$ \mathbb{Z}^d\to^D \mathbb{Z}^d \to^D \mathbb{Z}^d \dots $$
and calculating $\lim(\mathbb{Z}^d,D)$ is easy. This is rarely the case, however -- most of the time, $P^{-1}$ has rational, non-integral entries. Do I still get some connection between $\lim(\mathbb{Z}^d,A)$ and $\lim(\mathbb{Z}^d,D)$, or is it impossible to say anything in general?