  We present a result of localization for a matrix-valued Anderson-Bernoulli
operator, acting on $L^2(\R)\otimes \R^N$, for an arbitrary $N\geq 1$, whose
interaction potential is generic in the real symmetric matrices. For such a
generic real symmetric matrix, we construct an explicit interval of energies on
which we prove localization, in both spectral and dynamical senses, away from a
finite set of critical energies. This construction is based upon the formalism
of the F\"urstenberg group to which we apply a general criterion of density in
semisimple Lie groups. The algebraic nature of the objects we are considering
allows us to prove a generic result on the interaction potential and the
finiteness of the set of critical energies.
