In this talk we generalize well known results regarding minimal realizations of non-commutative (nc) rational functions, using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point.

In particular, we prove the existence and uniqueness of a minimal realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity. Moreover, we show that using this realization we can evaluate the function on all of its domain (of matrices of all sizes) and also w.r.t any stably finite algebras.

Unlike the case of a scalar centre, the coefficients of the realization can not be chosen arbitrarily. We present necessary and sufficient conditions (called the linearized lost-abbey conditions) on the coefficients of a minimal realization centred at a matrix point, such that there exists a nc rational function which admits the realization.

This is a joint work with Victor Vinnikov.