Recognizing cyclic matrices and a conjecture of J.G. Thompson

TL;DR

This paper proves Thompson's conjecture that any invertible matrix over a field can be permuted to become cyclic, by establishing a new criterion for cyclic matrices and applying it to the problem.

Contribution

It introduces a simple criterion for matrices to be cyclic and uses it to prove Thompson's long-standing conjecture.

Findings

Thompson's conjecture is proven true for all invertible matrices over a field.

A new criterion for identifying cyclic matrices is established.

The paper discusses an error in a related proof involving specific matrix properties.

Abstract

In 2006 J.G. Thompson conjectured: "If F is a field and A is in GL(n,F), then there is a permutation matrix P such that AP is cyclic, that is, the minimal polynomial of AP is also its characteristic polynomial" (open problem 16.95 in the Kourovka Notebook). The present note provides a simple criterion for a matrix to be cyclic and uses this to prove Thompson's conjecture. ERRATA I am indebted to Alexander Stasinski (Durham University) for the following observations. Suppose n > 2 and J is the n x n all 1's matrix over a field of characteristic not 2. Then A := J - I has the minimal polynomial (X + 1)(X - n + 1). Thus A is invertible and not cyclic even though A satisfies condition (iv) of the Proposition. The error lies in the claim towards the end of the proof that "these particular row and column errors do not change the determinants ...".