Companion Weakly Periodic Matrices over Finite and Countable Fields

TL;DR

This paper characterizes fields over which all companion matrices are weakly periodic of index 2, showing this occurs uniquely for certain countable fields of positive characteristic and finite fields of order greater than n.

Contribution

It provides a complete characterization of fields where all companion matrices are weakly periodic of index 2, extending previous results to broader classes of fields.

Findings

All companion matrices are weakly periodic of index 2 only over specific countable fields of positive characteristic.

In the commuting case, such fields are finite fields with order greater than n.

The results generalize earlier work by Breaz-Modoi (2016).

Abstract

We explore the situation where all companion $n \times n$ matrices over a field $F$ are weakly periodic of index of nilpotence $2$ and prove that this can be happen uniquely when $F$ is a countable field of positive characteristic, which is an algebraic extension of its minimal simple (finite) subfield, with all subfields of order greater than $n$. In particular, in the commuting case, we show even that $F$ is a finite field of order greater than $n$. Our obtained results somewhat generalize those obtained by Breaz-Modoi in Lin. Algebra & Appl. (2016).