In a previous paper the authors generalized classical results of minimal
realizations of non-commutative (nc) rational functions, using nc
Fornasini--Marchesini realizations which are centred at an arbitrary matrix
point. In particular, it was proved that the domain of regularity of a nc
rational function is contained in the invertibility set of a corresponding
pencil of any minimal realization of the function. In this paper we prove an
equality between the domain of a nc rational function and the domain of any of
its minimal realizations. As for evaluations over stably finite algebras, we
show that the domain of the realization w.r.t any such algebra coincides with
the so called matrix domain of the function w.r.t the algebra. As a corollary
we show that the domain of regularity and the stable extended domain coincide.
In contrary to both the classical case and the scalar case -- where every
matrix coefficients which satisfy the controllability and observability
conditions can appear in a minimal realization of a nc rational function -- the
matrix coefficients in our case have to satisfy certain equations, called
linearized lost-abbey conditions, which are related to Taylor--Taylor
expansions in nc function theory