Partial Smoothness of the Numerical Radius at Matrices Whose Fields of Values are Disks

Michael Overton, NYU Courant Institute
September 11, 2023 3:00 pm LSK 306

The field of values (numerical range) of a matrix A is the set of points in the complex plane of the form v*Av, where v is a vector with unit 2-norm, and the numerical radius of A is the maximum of the moduli of all such points. It is well known that the numerical radius is useful for modeling transient stability of discrete-time dynamical systems, via the “power inequality,” unlike the spectral radius, which models asymptotic stability. Optimization problems involving the numerical radius, in contrast to those involving the spectral radius, are typically tractable via semidefinite programming, and their solutions 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 M, with respect to which the numerical radius r is partly smooth, implying that r is smooth when restricted to M but strictly nonsmooth when restricted to lines transversal to M. Consequently, minimizers of the numerical radius of a parametrized matrix often lie in M. Partial smoothness holds, in particular, at n x 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 x 3 matrices, the disk matrices comprise the closure of a semi-algebraic manifold L with dimension 12, and the numerical radius is partly smooth with respect to L.

Refreshments will be served preceding the talk, starting at 2:30.