Matrix extreme points and free extreme points of free spectrahedra

Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form LA(X)=Id⊗In+A1⊗X1+A2⊗X2+⋯+Ag⊗Xg⪰0,where the Ai and Xi are symmetric matrices and the Xi have any size n×n. Free spectrahedra are ubiquitous in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are particularly important. We present theoretical, algorithmic, and experimental results illuminating basic properties of extreme points. For example, though many authors have studied matrix and free extreme points, it has until now been unknown if these two types of extreme points are actually different. This paper settles that issue.