Minimal degree coprime factorization of rational matrices

Given a rational matrix G with complex coefficients and a domain Gamma in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over Gamma, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the poles of G. We apply this result to obtain the parametrized class of all coprime factorizations over Gamma with denominators of minimal McMillan degree n(b)-the number of poles of G outside Gamma. Specific choices of the parameters and of Gamma allow us to determine coprime factorizations, as for instance, with polynomial, proper, or stable factors. Further, we consider the case in which the denominator has a certain symmetry, namely it is J all-pass with respect either to the imaginary axis or to the unit circle. We give necessary and sufficient solvability conditions for the problem of coprime factorization with J all-pass denominator of McMillan degree n(b) and, when a solution exists, we give a construction of the class of coprime factors. When no such solution exists, we discuss the existence of, and give solutions to, coprime factorizations with J all-pass denominators of minimal McMillan degree (> n(b)). All the developments are carried out in terms of descriptor realizations associated with rational matrices, leading to explicit and computationally efficient formulas.