Constructive solutions to spectral and inner–outer factorizations with respect to the disk

We use state-space realizations to solve the spectral and inner–outer factorization problems with respect to the unit disk formulated for a completely general rational matrix. The algorithm is based on orthogonal transformations and standard reliable procedures for solving Stein and Riccati equations. The formulas apply in particular to general descriptor systems or polynomial matrices. The main novelty is that we allow for arbitrary rank, poles and zeros on the unit circle, or at infinity.