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

Authors : Edoardo BALLICO

Pages : 1698-1710

View : 14 | Download : 4

Publication Date : 0000-00-00

Article Type : Research Paper

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 $\mathrm{Num} insert ignore into journalissuearticles values(M);$ of $M$ insert ignore into journalissuearticles values(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 $c\notin \mathrm{Num} insert ignore into journalissuearticles values(M);$. We compute $\mathrm{Num} insert ignore into journalissuearticles values(M);$ in some cases, mostly with $n=2$.
Keywords : Numerical range, sesquilinear form, formally real field, number field