  Hermitian linear matrix pencils are ubiquitous in control theory, operator
systems, semidefinite optimization, and real algebraic geometry. This survey
reviews the fundamental features of the matricial solution set of a linear
matrix inequality, the free spectrahedron, from the perspective of free real
algebraic geometry. Namely, among matricial solution sets of noncommutative
polynomial inequalities, free spectrahedra are precisely the convex ones.
Furthermore, a procedure for detecting free spectrahedra and producing their
representing linear matrix pencils is discussed. Finally, free spectrahedra
admit a perfect Positivstellensatz, leading to a semidefinite programming
formulation of eigenvalue optimization over convex matricial sets constrained
by noncommutative polynomial inequalities.
