Matrix extreme points and free extreme points of free spectrahedra.

Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form \[ L_A(X)=I_d{\otimes}I_n+A_1{\otimes}X_1+A_2{\otimes}X_2+\dots+A_g{\otimes}X_g\succeq 0, \]LA(X)=Id⊗In+A1⊗X1+A2⊗X2+⋯+Ag⊗Xg⪰0, where the $ A_i $ Ai and $ X_i $ Xi are symmetric matrices and the $ X_i $ Xi have any size $ n\times n $ 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.We also present and analyze several algorithms. Namely, we perfect an algorithm for computing an expansion of an element of a free spectrahedron in terms of free extreme points. We also give algorithms for testing if a point is matrix extreme and for computing matrix extreme points that are not free extreme. [ABSTRACT FROM AUTHOR]