  We present a result of absence of absolutely continuous spectrum in an
interval of $\R$, for a matrix-valued random Schr\"odinger operator, acting on
$L^2(\R)\otimes \R^N$ for an arbitrary $N\geq 1$, and whose interaction
potential is generic in the real symmetric matrices. For this purpose, we prove
the existence of an interval of energies on which we have separability and
positivity of the $N$ non-negative Lyapunov exponents of the operator. The
method, based upon the formalism of F\"urstenberg and a result of Lie group
theory due to Breuillard and Gelander, allows an explicit contruction of the
wanted interval of energies.
