Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary

TL;DR

This paper proves Gau and Wu's 2008 conjecture that 4-by-4 nilpotent matrices have at most two flat boundary portions in their numerical range, providing a detailed characterization of cases with two parallel flat portions.

Contribution

The paper confirms the conjecture and offers a comprehensive description of when two flat portions occur and are parallel, advancing understanding of numerical ranges of nilpotent matrices.

Findings

Maximum of two flat portions on boundary proven

Complete characterization of parallel flat portions provided

Additional independent results on numerical ranges included

Abstract

In their 2008 paper Gau and Wu conjectured that the numerical range of a 4-by-4 nilpotent matrix has at most two flat portions on its boundary. We prove this conjecture, establishing along the way some additional facts of independent interest. In particular, a full description of the case in which these two portions indeed materialize and are parallel to each other is included.