Pre-images of extreme points of the numerical range, and applications

Abstract

We extend the pre-image representation of exposed points of the numerical range of a matrix to all extreme points. With that we characterize extreme points which are multiply generated, having at least two linearly independent pre-images, as the extreme points which are Hausdorff limits of flat boundary portions on numerical ranges of a sequence converging to the given matrix. These studies address the inverse numerical range map and the maximum-entropy inference map which are continuous functions on the numerical range except possibly at certain multiply generated extreme points. This work also allows us to describe closures of subsets of 3-by-3 matrices having the same shape of the numerical range.

19 pages, no figures, v2 has corrected proofs of Lemma 2.2 and Lemma 3.3

References in corpus (3)

• Information-theoretical analysis of topological entanglement entropy and multipartite correlations
• Closures of exponential families
• Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

Cited by in corpus (3)

• Classification of joint numerical ranges of three hermitian matrices of size three
• A new signature of quantum phase transitions from the numerical range
• Matrix systems, algebras, and open maps