  We show that, in 't Hooft's large N limit, matrix models can be formulated as
a classical theory whose equations of motion are the factorized
Schwinger--Dyson equations. We discover an action principle for this classical
theory. This action contains a universal term describing the entropy of the
non-commutative probability distributions. We show that this entropy is a
nontrivial 1-cocycle of the non-commutative analogue of the diffeomorphism
group and derive an explicit formula for it. The action principle allows us to
solve matrix models using novel variational approximation methods; in the
simple cases where comparisons with other methods are possible, we get
reasonable agreement.
