Existence of minimal nonsquare J-symmetric factorizations for self-adjoint rational matrix functions

In this paper we show that any rational matrix function having hermitian values on the imaginary axis, and with constant signature and constant pole signature admits a minimal symmetric factorization with possibly nonsquare factors. Our proof is based on a construction which shows that any such function can be extended (preserving its McMillan degree) to a function that admits J-symmetric factorization with square factors. Also, we consider other properties of the factors in J-symmetric factorizations. Particular attention is given to the study of the common invariant zero structure of these factors.