A numerical range characterization of unitary matrices over a finite field.

Let q be a prime power, q ≠ 2 , 3 , and let A be an n × n matrix over q 2 . We prove that the numerical range of A B and B A is the same for every rank 1 matrix B if and only if A 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]. [ABSTRACT FROM AUTHOR]