An introduction to matrix convex sets and free spectrahedra

The purpose of this paper is to give a self-contained overview of the theory of matrix convex sets and free spectrahedra. We will give new proofs and generalizations of key theorems. However we will also introduce various new concepts and results as well. Key contributions of this paper are:

• A new general Krein-Milman theorem that characterizes the smallest operator tuple defining a compact matrix convex set.
• The introduction and a characterization of matrix exposed points.
• A (weak) Minkowski theorem in the language of matrix extreme points (with a new proof of the weak Krein-Milman theorem of Webster and Winkler).
• Simplified/new proofs of the Gleichstellensatz, Helton and McCulloughs characterization of free spectrahedra as closures of matrix convex “free basic open semialgebraic” sets and a characterization of hermitian irreducible free loci of Helton, Klep and Vol$\check{\text{c}}$i$\check{\text{c}}$.