  We consider products of random matrices that are small, independent
identically distributed perturbations of a fixed matrix $T_0$. Focusing on the
eigenvalues of $T_0$ of a particular size we obtain a limit to a SDE in a
critical scaling. Previous results required $T_0$ to be a (conjugated) unitary
matrix so it could not have eigenvalues of different modulus. From the result
we can also obtain a limit SDE for the Markov process given by the action of
the random products on the flag manifold. Applying the result to random
Schr\"odinger operators we can improve some result by Valko and Virag showing
GOE statistics for the rescaled eigenvalue process of a sequence of Anderson
models on long boxes. In particular we solve a problem posed in their work.
