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

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 .