Convexity of the joint numerical range: topological and differential geometric viewpoints

By E. Gutkin, E. Jonckheere, M. Karow. 2004

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Abstract We investigate the convexity of the joint numerical range of m -tuples of n × n hermitian matrices. The methods come from differential geometry and the differential and algebraic topology. Our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n . Modulo a mild technical assumption, this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the ( m −1)-dimensional unit sphere. The condition is that the eigenvalue has constant multiplicity. We show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges. As a byproduct of our main result, we obtain a new proof of the celebrated Toeplitz–Hausdorff theorem, and of its extension: The numerical range of any triple of hermitian n × n matrices is convex if n >2.

Subfields: Astronomy and Astrophysics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9f91530668368f18d65bc582f04d90b61388232e

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In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

A Survey of the S-Lemma (2007, background): “, Bm)? Most of the results (see [7, 32, 49]) on the convexity of the numerical range investigate the question over the complex field.”
Geometry of Quantum States: Index (2007, background): “In general, for L = 3 and N ≥ 3 this set is convex [597], while for L ≥ 4 matrices of size N ≥ 3 this is not the case [384].”