Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting

We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini–Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.