In this paper, we consider an arbitrary matrix-valued, rational spectral density Phi(z). We show with a constructive proof that Phi(z) admits a factorization of the form Phi(z)=W-T (z(-1)) W(z), where W(z) is stochastically minimal. Moreover, W(z) and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic-type conditions. By suitably selecting the analyticity regions, this extremely general result particularizes into a corollary that may be viewed as the discrete-time counterpart of the matrix factorization method devised by Youla in his celebrated work [48].