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

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 ...".