We consider a function-analytic approach to study synchronizing automata, primitive and ergodic matrix families. This gives a new way to establish some criteria for primitivity and for ergodicity of families of nonnegative matrices. We introduce a concept of canonical partition and use it to construct a polynomial-time algorithm for finding a positive matrix product and an ergodic matrix product whenever they exist. This also provides a generalization of some results of the Perron-Frobenius theory from one nonnegative matrix to families of matrices. Then we define the h-synchronizing automata and prove that the existence of a reset tuple is polynomially decidable. The question whether the functional-analytic approach can be extended to the h-primitivity is addressed and several open problems are formulated.

我们考虑一种功能分析方法来研究同步自动机，原始和遍历矩阵族。这提供了一种新方法，可以为原始性和非负矩阵族的遍历性建立一些标准。我们引入规范划分的概念，并用其构造多项式时间算法，以在存在正矩阵乘积和遍历矩阵乘积时找到它们。这也提供了Perron-Frobenius理论的一些结果的概括，从一个非负矩阵到矩阵族。然后，我们定义h同步自动机，并证明重置元组的存在是多项式可判定的。功能分析方法是否可以扩展到h的问题-解决了原始性问题，并提出了一些未解决的问题。