  For a principal ideal domain $A$, the Latimer--MacDuffee correspondence sets
up a bijection between the similarity classes of matrices in
$\operatorname{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and
the ideal classes of the order $A[x]/(f(x))$. We prove that when $A[x]/(f(x))$
is maximal (i.e., integrally closed, i.e., a Dedekind domain), then every
similarity class contains a representative that is, in a sense, close to being
a companion matrix. The first step in the proof is to show that any similarity
class corresponding to an ideal (not necessarily prime) of degree one contains
a representative of the desired form. The second step is a previously
unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is
maximal, every ideal class contains an ideal of degree one.
