In this paper we discuss the mathematical background and the computational aspects which underly the implementation of a collection of Julia functions in the MatrixPencils package for the determination of structural properties of polynomial and rational matrices. We primarily focus on the computation of the finite and infinite spectral structures (e.g., eigenvalues, zeros, poles) as well as the left and right singular structures (e.g., Kronecker indices), which play a fundamental role in the structure of the solution of many problems involving polynomial and rational matrices. The basic analysis tool is the determination of the Kronecker structure of linear matrix pencils using numerically reliable algorithms, which is used in conjunction with several linearization techniques of polynomial and rational matrices. Examples of polynomial and rational matrices, which exhibit all relevant structural features, are considered to illustrate the main mathematical concepts and the capabilities of implemented tools.

中文翻译：

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使用 Julia 计算多项式和有理矩阵的克罗内克结构

在本文中，我们讨论了数学背景和计算方面，它们是在 MatrixPencils 包中实现一组 Julia 函数的基础，用于确定多项式和有理矩阵的结构特性。我们主要关注有限和无限谱结构（例如，特征值、零点、极点）以及左右奇异结构（例如，克罗内克指数）的计算，它们在解的结构中起着基本作用许多涉及多项式和有理矩阵的问题。基本分析工具是使用数值可靠的算法确定线性矩阵铅笔的 Kronecker 结构，该算法与多项式和有理矩阵的几种线性化技术结合使用。