Kippenhahn's construction revisited

TL;DR

This paper revisits Kippenhahn's construction of the numerical range of matrices, providing a counterexample that challenges the original assertion and clarifies the geometric and algebraic aspects involved.

Contribution

It presents a counterexample to Kippenhahn's construction, prompting a detailed review and clarification of the proof using convex and real algebraic geometry methods.

Findings

Counterexample shows the construction fails at singular points.

Clarification of Kippenhahn's proof with geometric methods.

Highlights limitations of the original algebraic approach.

Abstract

Kippenhahn discovered that the numerical range of a complex square matrix is the convex hull of a plane real algebraic curve. Here, we present an example of a convex set, which has a similar algebraic description as the numerical range, whereas the analogue of Kippenhahn's construction fails regarding isolated, singular points of the curve. This example prompted us to carefully review Kippenhahn's assertion and to highlight aspects of a complete proof that was achieved with methods of convex geometry and real algebraic geometry.