  We give a non-perturbative definition of U(n) gauge theory on fuzzy CP^2 as a
multi-matrix model. The degrees of freedom are 8 hermitian matrices of finite
size, 4 of which are tangential gauge fields and 4 are auxiliary variables. The
model depends on a noncommutativity parameter 1/N, and reduces to the usual
U(n) Yang-Mills action on the 4-dimensional classical CP^2 in the limit N ->
\infty. We explicitly find the monopole solutions, and also certain U(2)
instanton solutions for finite N. The quantization of the model is defined in
terms of a path integral, which is manifestly finite. An alternative
formulation with constraints is also given, and a scaling limit as R^4_\theta
is discussed.
