Cyclic matrices and polynomial interpolation over division rings

Abstract

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.