Partial smoothness of the numerical radius at matrices whose fields of values are disks
Adrian S. Lewis, Michael L. Overton
TL;DR
The paper analyzes the geometry of the numerical radius around disk matrices, i.e., matrices whose field of values $W(A)$ is a disk centered at the origin. It develops a variational framework based on subdifferentials and partial smoothness, providing an explicit description of $\partial r(X)$ for disk matrices and identifying when disk matrices form analytic manifolds of codimension $2n$ (and, in ${\bf M}^3$, a 12-dimensional semialgebraic set inside the 18-dimensional space). The theory is applied to canonical sparse patterns, including 2×2 Jordan blocks, the Crabb matrix, and general superdiagonal forms, with detailed treatment of the 3×3 case. The results shed light on why optimization with the numerical radius frequently yields disk matrices and have implications for proximal methods and sensitivity analysis in matrix optimization, while leaving open questions about broader SDP characterizations and higher-dimensional structures.
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.