A numerical range characterization of unitary matrices over a finite field

Abstract

Let [Formula: see text] be a prime power, [Formula: see text], and let [Formula: see text] be an [Formula: see text] matrix over [Formula: see text]. We prove that the numerical range of [Formula: see text] and [Formula: see text] is the same for every rank [Formula: see text] matrix [Formula: see text] if and only if [Formula: see text] is a multiple of a unitary matrix. The corresponding result for complex matrices was proved by Chien, Gau, Li, Tsai and Wang [Products of operators and numerical range, Linear Multilinear Algebra 64(1) (2016) 58–67].