Coval description of the boundary of a numerical range and the secondary values of a matrix

TL;DR

This paper introduces a new coval description of the boundary of a matrix's numerical range using distances to tangent lines and secondary values, revealing simpler geometric properties than traditional eigenvalue-based descriptions.

Contribution

It presents a novel coval approach to describe the numerical range boundary using distances to tangent lines and secondary values, simplifying the algebraic complexity.

Findings

Boundary curves are algebraic, closed, and simple.

Distances to tangent lines and secondary values characterize the boundary.

Secondary values have well-defined algebraic and geometric properties.

Abstract

The boundary of a numerical range of a finite matrix is always a nice curve (algebraic, closed and simple), but the equation it satisfies is often very complicated. We will show that, furthermore, there is no hope of describing these curves in terms of distances from the eigenvalues -- as the dimension~2, where the numerical range is just an ellipse, would suggest. But, as we will show, there is a remarkably simple ``coval'' description in terms of distances to \textit{tangent lines}. Provided that one measures these distances not only to the eigenvalues but also to additional points, the most important of which are \textit{secondary values} -- which we will define and describe their algebraic and geometric properties.