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Gradient Flows Computing the C-numerical Range with Applications in NMR Spectroscopy |
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| Authors: | U Helmke K Hüper JB Moore Th Schulte-Herbrüggen |
| Institution: | (1) Department of Mathematics, University of Würzburg, Am Hubland, D-97074 Würzburg, Germany;(2) Department of Systems Engineering, RSISE, The Australian National University, Canberra, ACT, 0200, Australia;(3) Department of Chemistry, Carlsberg Laboratory, DK-2500 Valby, Denmark |
| Abstract: | 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. |
| Keywords: | C-numerical range Optimization Gradient flows Discretization NMR spectroscopy Nilpotent matrices |
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