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 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 F into a controllable or an observable pair. Using the characterization of ideals in F[z] in terms of controllable and observable pairs we consider ideal interpolation schemes in F[z] which merge into polynomial interpolation problems containing both left and right interpolation conditions.