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

TL;DR

This paper extends the concept of the numerical range to matrices over degree 2 Galois field extensions, highlighting key differences from classical and finite field cases, and computes this range in specific instances.

Contribution

It introduces a new definition of the numerical range for matrices over degree 2 Galois extensions and explores its properties, contrasting with classical and finite field scenarios.

Findings

Numerical range can include eigenvalues not in the set for certain fields.

Differences arise when the norm map's image is not closed under addition.

Explicit computations of the numerical range are provided for 2x2 matrices.

Abstract

Let $L$ be a degree $2$ Galois extension of the field $K$ and $M$ an $n\times n$ matrix with coefficients in $L$. Let $\langle \ ,\ \rangle : L^n\times L^n\to L$ be the sesquilinear form associated to the involution $L\to L$ fixing $K$. We use $\langle \ ,\ \rangle$ to define the numerical range $\mathrm{Num} (M)$ of $M$ (a subset of $L$), extending the classical case $K=\mathbb {R}$, $L=\mathbb {C}$ and the case of a finite field introduced by Coons, Jenkins, Knowles, Luke and Rault. There are big differences with respect to both cases for number fields and for all fields in which the image of the norm map $L\to K$ is not closed by addition, e.g., $c\in L$ may be an eigenvalue of $M$, but $c\notin \mathrm{Num} (M)$. We compute $\mathrm{Num} (M)$ in some case, mostly with $n=2$.