Cyclic matrices and polynomial interpolation over division rings

TL;DR

This paper explores canonical forms of cyclic matrices over division rings by embedding them into controllable or observable pairs, and investigates polynomial interpolation schemes involving both left and right conditions.

Contribution

It introduces a new approach to canonical forms of cyclic matrices over division rings using controllable and observable pairs, and studies ideal interpolation schemes with mixed conditions.

Findings

Characterization of ideals in $\

Development of polynomial interpolation problems with combined left and right conditions.

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.