The space of probability measures endowed with the optimal transport metric has a rich structure with applications in probability, analysis, and geometry. The notion of (displacement) convexity in this space was discovered by McCann, and forms the backbone of this theory. I will introduce a new, and stronger, notion of displacement convexity which operates on the matrix level. The motivation behind this definition is to capture the intrinsic dimensionality of probability measures which could have very different behaviors along different directions in space. I will show that a broad class of flows satisfy matrix displacement convexity: heat flow, optimal transport, entropic interpolation, mean-field games, and semiclassical limits of non-linear Schrödinger equations. This leads to intrinsic dimensional functional inequalities which provide a systematic improvement on numerous classical functional inequalities.