  A spectrahedron is a convex set defined by a linear matrix inequality, i.e.,
the set of all $x \in \mathbb{R}^g$ such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2
+ \dots + A_g x_g \succeq 0 \] for some symmetric matrices $A_1,\ldots,A_g$.
This can be extended to matrix spaces by taking $X$ to be a tuple of real
symmetric matrices of any size and using the Kronecker product $$L_A(X) = I_n
\otimes I_d + A_1 \otimes X_1 + A_2 \otimes X_2 + \dots + A_g \otimes X_g.$$
The solution set of $L_A (X) \succeq 0$ is called a \textit{free
spectrahedron}. Free spectrahedra are important in systems engineering,
operator algebras, and the theory of matrix convex sets. Matrix and free
extreme points of free spectrahedra are of particular interest. While many
authors have studied matrix and free extreme points of free spectrahedra, it
has until now been unknown if these two types of extreme points are actually
different. The results of this paper fall into three categories: theoretical,
algorithmic, and experimental. Firstly, we prove the existence of matrix
extreme points of free spectrahedra that are not free extreme. This is done by
producing exact examples of matrix extreme points that are not free extreme. We
also show that if the $A_i$ are $2 \times 2$ matrices, then matrix and free
extreme points coincide. Secondly, we detail methods for constructing matrix
extreme points of free spectrahedra that are not free extreme, both exactly and
numerically. We also show how a recent result due to Kriel (Complex
Anal.~Oper.~Theory 2019) can be used to efficiently test whether a point is
matrix extreme. Thirdly, we provide evidence that a substantial number of
matrix extreme points of free spectrahedra are not free extreme. Numerical work
in another direction shows how to effectively write a given tuple in a free
spectrahedron as a matrix convex combination of its free extreme points.
