Positive semidefinite rank and nested spectrahedra

TL;DR

This paper explores the geometric structure of matrices with a fixed positive semidefinite rank, describing their boundaries through spectrahedral nesting and proposing conjectures for general cases.

Contribution

It introduces a geometric approach to understanding the boundary of matrices with fixed positive semidefinite rank, including conjectures for general cases.

Findings

Boundary described for small cases

Conjecture for general boundary structure

Nesting spectrahedra as a key technique

Abstract

The set of matrices of given positive semidefinite rank is semialgebraic. In this paper we study the geometry of this set, and in small cases we describe its boundary. For general values of positive semidefinite rank we provide a conjecture for the description of this boundary. Our proof techniques are geometric in nature and rely on nesting spectrahedra between polytopes.