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

Authors

EDOARDO BALLICO

DOI

10.3906/mat-1711-103

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 cases, mostly with $n=2$.

Keywords

Numerical range, sesquilinear form, formally real field, number field

First Page

1698

Last Page

1710

Recommended Citation

BALLICO, EDOARDO (2018) "On the numerical range of square matrices with coefficients in a degree $2$ Galois field extension," Turkish Journal of Mathematics: Vol. 42: No. 4, Article 12. https://doi.org/10.3906/mat-1711-103
Available at: https://journals.tubitak.gov.tr/math/vol42/iss4/12