  We here specialize the well known Positive Real Lemma (also known as the
Kalman-Yakubovich-Popov Lemma) to complex matrix-valued rational functions, (i)
generalized positive even and (ii) odd. On the way we characterize the (non)
minimality of realization of arbitrary systems through (i) the corresponding
state matrix and (ii) moving the poles by applying static output feedback. We
then explore the application of static output feedback to both generalized
positive even and to odd functions.
