We prove a full large deviations principle, in the scale N2,
for the empirical measure of the eigenvalues of an N x N
(non self-adjoint) matrix composed of i.i.d. zero mean random
variables with variance N-1. The (good) rate function which
governs this rate function possesses as unique minimizer the
circular law, providing an alternative proof of convergence to
the latter. The techniques are related to recent work by Ben
Arous and Guionnet, who treat the self-adjoint case. A crucial
role is played by precise determinant computations due to Edelman
and to Lehmann and Sommers.