Realizations of Non-commutative Rational Functions Around a Matrix Centre, II: The Lost-Abbey Conditions

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