A proof of the elliptical range theorem via Kippenhahn's theorem

TL;DR

This paper provides a simple proof of the elliptical range theorem for 2x2 matrices using properties of plane algebraic curves and Kippenhahn's theorem, simplifying previous complex proofs.

Contribution

It introduces a straightforward proof of the elliptical range theorem based on algebraic geometry and Kippenhahn's theorem, avoiding complex computations.

Findings

The field of values of a 2x2 matrix is an elliptical disk.

The proof leverages properties of plane algebraic curves.

Kippenhahn's theorem underpins the proof.

Abstract

The elliptical range theorem asserts that the field of values (or numerical range) of a two-by-two matrix with complex entries is an elliptical disk, the foci of which are the eigenvalues of the given matrix. Many proofs of this result are available in the literature, but most, with one exception, are computational and quite involved. In this note, it is shown that the elliptical range theorem follows from the properties of plane algebraic curves and a straightforward application of a well-known result due to Kippenhahn.