  A new notion of displacement convexity on a matrix level is developed for
density flows arising from mean-field games, compressible Euler equations,
entropic interpolation, and semi-classical limits of non-linear Schr\"odinger
equations. Matrix displacement convexity is stronger than the classical notions
of displacement convexity, and its verification (formal and rigorous) relies on
matrix differential inequalities along the density flows. The matrical nature
of these differential inequalities upgrades dimensional functional inequalities
to their intrinsic dimensional counterparts, thus improving on many classical
results. Applications include turnpike properties, evolution variational
inequalities, and entropy growth bounds, which capture the behavior of the
density flows along different directions in space.
