# Partial smoothness of the numerical radius at matrices whose fields of values are disks

## TL;DR

This paper investigates the partial smoothness of the numerical radius at matrices with disk-shaped fields of values, revealing geometric structures and applications to specific matrix classes.

## Contribution

It characterizes the manifold structure of disk matrices where the numerical radius is partly smooth, and applies these findings to special matrix classes like Jordan blocks.

## Key findings

- Disk matrices form a semi-algebraic manifold of dimension 12 in 18-dimensional space.
- The set of disk matrices exhibits partial smoothness for the numerical radius.
- Results extend to matrices with a single superdiagonal and complex 3x3 matrices.

## Abstract

Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of "disk matrices" is a manifold with respect to which the numerical radius is partly smooth. We then apply our results to matrices whose nonzeros consist of a single superdiagonal, such as Jordan blocks and the Crabb matrix related to a well-known conjecture of Crouzeix. Finally, we consider arbitrary complex three-by-three matrices; even in this case, the details are surprisingly intricate. One of our results is that in this real vector space with dimension 18, the set of disk matrices is a semi-algebraic manifold with dimension 12.