论文标题
矩阵中心周围的非交通合理功能的实现，ii：丢失的条件
Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions
论文作者
论文摘要
在先前的论文中，作者使用NC Fornasini-Marchesini实现的最小实现（NC）合理函数的最小实现经典结果，这些实现位于任意矩阵点。特别是，事实证明，NC合理函数的规律性域包含在该函数的任何最小实现的相应铅笔的可逆性集中。在本文中，我们证明了NC合理函数的领域与其最小实现的领域之间的平等性。至于对稳定有限代数的评估，我们表明实现W.R.T的结构域与代数W.R.T函数的所谓矩阵域重合。作为推论，我们表明规律性和稳定的扩展域的域重合。与经典案例和标量案例相反 - 满足可控性和可观察性条件的每个矩阵系数可以在最小化NC合理功能的最小化中出现 - 在我们的情况下，矩阵系数必须满足某些方程式，必须满足与taylor-taylor-taylor-taylor-taylor-taylor-taylor-taylor-taylor-taylor-taylor-taylor-taylor-tay-bebey条件相关。
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.
