All doubly coprime factorizations of a general rational matrix

For a general rational matrix function (possibly polynomial or improper) we give state-space formulas for the class of all doubly coprime factorizations over an arbitrary fixed set in the closed complex plane. The main result is expressed in terms of a special type of realization–called centered–that exhibits the same attractive features and allows for formulas that bear essentially the same elegant simplicity of the proper case. In particular, the formulas can be specialized to get the parametrization of all proper, polynomial, or proper and stable doubly coprime factorizations of an arbitrary descriptor system. En route, we provide a closed state-space formula for the inverse of an invertible general rational matrix.