In this paper, we investigate the numerical range of matrices over finite fields, particularly triangular matrices. We conjecture that all strictly triangular matrices over finite fields of dimension 3 or greater have a numerical range encompassing the entirety of the finite field. We use both algebraic and computational methods to support this claim, making some concrete progress towards the algebraic proof. Further, we conjecture that all matrices over finite fields have a numerical range falling into one of five potential categories, providing an extensive appendix of randomly generated computational examples which seems to support this conjecture.
Russell, Ariel, "Numerical Range of Strictly Triangular Matrices Over Finite Fields" (2023). Mathematics Student Projects. 4.