Spring 2019

Kippenhahn's Theorem in Higher Dimensions

Friday, May 3, 2019 11:15 am12:00 pm PDT

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Daniel Plaumann (TU Dortmund University)

By the Toeplitz-Hausdorff theorem in convex analysis, the numerical range of a complex square matrix is a convex compact subset of the complex plane. Kippenhahn's theorem describes the numerical range as the convex hull of an algebraic curve that is dual to a hyperbolic curve. For the joint numerical range of several matrices, the direct analogue of Kippenhahn's theorem is known to fail. We discuss the geometry behind these results and prove a generalization of Kippenhahn's theorem that holds in any dimension.

Joint work with Rainer Sinn and Stephan Weis.