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

TL;DR

This paper extends the understanding of extreme points in the numerical range of matrices, characterizing their pre-images and limits, with implications for inverse numerical range and maximum-entropy inference maps.

Contribution

It generalizes pre-image representations to all extreme points and characterizes multiply generated points as limits of boundary portions, advancing inverse map analysis.

Findings

Extended pre-image representation to all extreme points.

Characterized multiply generated extreme points as limits of boundary portions.

Described closures of subsets of 3x3 matrices with similar numerical range shapes.

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.