On the numerical range of square matrices with coefficients in a degree 2 Galois field extension

Abstract

Let L be a degree 2 Galois extension of the field K and M an n × n matrix with coefficients in L. Let⟨ , ⟩ : Ln × Ln → L be the sesquilinear form associated to the involution L → L fixing K . We use ⟨ , ⟩ to define thenumerical range Num(M) of M (a subset of L), extending the classical case K = R, L = C, and the case of a finitefield introduced by Coons, Jenkins, Knowles, Luke, and Rault. There are big differences with respect to both cases fornumber fields and for all fields in which the image of the norm map L → K is not closed by addition, e.g., c ∈ L maybe an eigenvalue of M , but c ∈/ Num(M). We compute Num(M) in some cases, mostly with n = 2.