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For a principal ideal domain $A$, the Latimer--MacDuffee correspondence setsup a bijection between the similarity classes of matrices in$\operatorname{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ andthe 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 everysimilarity class contains a representative that is, in a sense, close to beinga companion matrix. The first step in the proof is to show that any similarityclass corresponding to an ideal (not necessarily prime) of degree one containsa representative of the desired form. The second step is a previouslyunpublished result due to Lenstra that implies that when $A[x]/(f(x))$ ismaximal, every ideal class contains an ideal of degree one.

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes