Consider an m-dimensional analytic cocycle with underlying dynamics given by an irrational translation on the circle. Assuming that the d-dimensional upper left corner of the cocycle is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrodinger operators. [This is joint work with Pedro Duarte.]