Gradient Flows Computing the C-numerical Range with Applications in NMR Spectroscopy

In this paper gradient flows on unitary matrices are studied that maximize the real part of the C-numerical range of an arbitrary complex n × n-matrix A. The geometry of the C-numerical range can be quite complicated and is only partially understood. A numerical discretization scheme of the gradient flow is presented that converges to the set of critical points of the cost function. Special emphasis is taken on a situation arising in NMR spectroscopy where the matrices C, A are nilpotent and the C-numerical range is a circular disk in the complex plane around the origin.