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.
Keywords
Cite
@article{arxiv.1509.05676,
title = {Pre-images of extreme points of the numerical range, and applications},
author = {Ilya M. Spitkovsky and Stephan Weis},
journal= {arXiv preprint arXiv:1509.05676},
year = {2020}
}
Comments
19 pages, no figures, v2 has corrected proofs of Lemma 2.2 and Lemma 3.3