Partial Smoothness of the Numerical Radius at Matrices Whose Fields of Values are Disks
Adrian S. Lewis, Michael L. Overton · SIAM Journal on Matrix Analysis and Applications · 2020
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 $\mathcal M$, with respect to which the numerical radius $r$ is partly smooth, implying that $r$ is smooth when restricted to $\mathcal M$ but strictly nonsmooth when restricted to lines transversal to $\mathcal M$. Consequently, minimizers of the numerical radius of a parametrized matrix often lie in $\mathcal M$. 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 $\mathcal L$ with dimension 12, and the numerical radius is partly smooth with respect to $\mathcal L$.