Semidefinite geometry of the numerical range

TL;DR

This paper explores the geometric structure of the numerical range of matrices using semidefinite cone theory, revealing dualities and algebraic connections with implications for semidefinite programming and statistical quadratic forms.

Contribution

It establishes a geometric duality between the numerical range and affine sections of the semidefinite cone, linking algebraic, geometric, and optimization perspectives.

Findings

Numerical range is an affine projection of the semidefinite cone.

Feasible sets of 2D LMIs are dual to numerical ranges.

Connections between algebraic curves and convex hulls are demonstrated.

Abstract

The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite cone. Both primal and dual sets can also be viewed as convex hulls of explicit algebraic plane curve components. Several numerical examples illustrate this interplay between algebra, geometry and semidefinite programming duality. Finally, these techniques are used to revisit a theorem in statistics on the independence of quadratic forms in a normally distributed vector.