Many ideas from convex analysis and real algebraic geometry extend canonically to the operator space setting giving rise to the notions of matrix (non-commutative) convex sets and functions. These notions also model matrix inequalities which are scalable in the sense that they do not explicitly depend upon the size of the matrices involved. This talk will survey matrix convexity emphasizing the rigid nature of convexity in the non-commutative semi-algebraic setting. It may aslo include a discussion of characterizing factorizations of a non-commutative polynomial in terms of the signature of its Hessian.