In this paper we generalize classical results regarding minimal realizations of non-commutative (nc) rational functions using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. 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 with respect to any stably finite algebra. As a corollary we obtain a new proof of the theorem by Cohn and Amitsur, that equivalence of two rational expressions over matrices implies that the expressions are equivalent over all stably finite algebras. Applications to the matrix valued and the symmetric cases are presented as well.

中文翻译：

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围绕矩阵中心的非交换有理函数的实现，I：稳定有限代数的综合、最小实现和评估

在本文中，我们使用以任意矩阵点为中心的 nc Fornasini-Marchesini 实现来概括关于非交换 (nc) 有理函数的最小实现的经典结果。我们证明了每个 nc 有理函数的最小实现的存在性和唯一性，其中心位于其正则域中的任意矩阵点。此外，我们表明，使用这种实现，我们可以在其所有域（所有大小的矩阵）上以及关于任何稳定有限代数。作为推论，我们获得了 Cohn 和 Amitsur 定理的新证明，矩阵上两个有理表达式的等价意味着这些表达式在所有稳定有限代数上是等价的。还介绍了对矩阵取值和对称情况的应用。