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

A. S. Lewis, M. L. Overton

Research output: Contribution to journalArticlepeer-review

## Abstract

Solutions to optimization problems involving the numerical radius often belong to the class of disk matrices"": those whose field of values is a circular disk in the complex plane centered at zero. We investigate this phenomenon using the variational-analytic idea of partial smoothness. We give conditions under which the set of disk matrices is locally a manifold \scrM , with respect to which the numerical radius r is partly smooth, implying that r is smooth when restricted to \scrM but strictly nonsmooth when restricted to lines transversal to \scrM . Consequently, minimizers of the numerical radius of a parametrized matrix often lie in \scrM . Partial smoothness holds, in particular, at n \times n matrices with exactly n - 1 nonzeros, all on the superdiagonal. On the other hand, in the real 18-dimensional vector space of complex 3 \times 3 matrices, the disk matrices comprise the closure of a semialgebraic manifold \scrL with dimension 12, and the numerical radius is partly smooth with respect to \scrL .

Original language English (US) 1004-1032 29 SIAM Journal on Matrix Analysis and Applications 41 3 https://doi.org/10.1137/18M1236289 Published - Jul 2020

## Keywords

• Crouzeix's conjecture