Numerical range over finite fields: Restriction to subspaces

Abstract

Let q be a prime power. For u=(u1,…,un),v=(v1,…,vn)∈Fq2n let 〈u,v〉:=∑i=1nuiqvi be the Hermitian form of Fq2n. For any n×n matrix M over Fq2 set Numk(M):={〈u,Mu〉|u∈Fq2n,〈u,u〉=k}. Num1(M) is the numerical range of q. In this paper we use Num1 to give a characterization of hermitian matrices, give some examples of 3×3 matrices M with Numk(M)=Fq2 for all k∈Fq and study Fq and Fq2 linear subspaces V⊂Mn,n(Fq2) such that either ♯(Num1(M))≤q for all M∈V or ♯(Num1(M))>q for all M∈V∖{0}.