Minimal indices cancellation and rank revealing factorizations for rational matrix functions

We study the problem of eliminating the minimal indices (the indices of a minimal polynomial basis of the null space) of a rational matrix function by multiplication with a suitably chosen invertible rational matrix function. We derive the class of all invertible factors that dislocate the minimal indices to certain zero locations and feature minimal McMillan degree. We impose additional conditions on the factor like being J-unitary, or J-inner, either with respect to the imaginary axis or to the unit circle, and characterize the classes of solutions. En route we extend the well-known rank revealing factorization of a constant matrix to rational matrix functions. The results are completely general and apply in particular to matrices which are polynomial, strictly proper or improper, rank deficient, with arbitrary poles and zeros including at infinity. All characterization are made by using descriptor realizations while the associated computations are performed by employing (constant) unitary transformations and standard reliable procedures for eigenvalue assignment or for solving Riccati equations.