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In a previous paper the authors generalized classical results of minimalrealizations of non-commutative (nc) rational functions, using ncFornasini--Marchesini realizations which are centred at an arbitrary matrixpoint. In particular, it was proved that the domain of regularity of a ncrational function is contained in the invertibility set of a correspondingpencil of any minimal realization of the function. In this paper we prove anequality between the domain of a nc rational function and the domain of any ofits minimal realizations. As for evaluations over stably finite algebras, weshow that the domain of the realization w.r.t any such algebra coincides withthe so called matrix domain of the function w.r.t the algebra. As a corollarywe show that the domain of regularity and the stable extended domain coincide.In contrary to both the classical case and the scalar case -- where everymatrix coefficients which satisfy the controllability and observabilityconditions can appear in a minimal realization of a nc rational function -- thematrix coefficients in our case have to satisfy certain equations, calledlinearized lost-abbey conditions, which are related to Taylor--Taylorexpansions in nc function theory.

Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions