  As is well known, any complex cyclic matrix $A$ is similar to the unique
companion matrix associated with the minimal polynomial of $A$. On the other
hand, a cyclic matrix over a division ring $\mathbb F$ is similar to a
companion matrix of a polynomial which is defined up to polynomial similarity.
In this paper we study more rigid canonical forms by embedding a given cyclic
matrix over a division ring $\mathbb F$ into a controllable or an observable
pair. Using the characterization of ideals in $\mathbb F[z]$ in terms of
controllable and observable pairs we consider ideal interpolation schemes in
$\mathbb F[z]$ which merge into a polynomial interpolation problems containing
both left and right interpolation conditions.
