Minimal cascade factorizations of real and complex rational matrices

A cascade factorization R ()= RI (,). R 2(x) .. () of an nXn nonsingular rational matrix R( X) is said to be minimal when the McMillan degrees of the factors add up to the McMillan degree of the original matrix. In this paper we give necessary and sufficient conditions for such a factorization to exist in terms of a state-space realization for R (2X). Next, we focus on numerical and algorithmic aspects. We discuss the numerical conditioning of the problem and we give algorithms to compute degree one factorizations and real degree two factorizations. Finally, we discuss the special case where R(X) is a para J-unitary matrix.