SIAM Review, Volume 62, Issue 1, Page 264-280, January 2020.
In this work we introduce a new operator based approach to matrix analysis. Our main technical tool comprises an extension of a tool introduced long ago by MacMahon to analyze the partitions of natural numbers: the Omega operator calculus. More precisely, we construct an operator acting linearly on absolutely convergent matrix valued expansions which selects appropriate terms of those expansions. In the context of our framework a new representation of matrix valued functions is available. Our representation is simple, requiring only the computation of matrix inverses and basic manipulations of the Taylor series of scalar functions. To show the usefulness of our approach we obtain fundamental results related to the basic theory of ODEs, perturbative calculations, multiple integrals involving the matrix exponential, the Sylvester equation, the multivariate Faà di Bruno formula, Hermite polynomials, queuing theory, and graph theory.

中文翻译：

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矩阵分析和欧米茄微积分

SIAM评论，第62卷，第1期，第264-280页，2020年1月。
在这项工作中，我们介绍了一种基于算子的新方法来进行矩阵分析。我们的主要技术工具包括MacMahon早就引入的一种工具的扩展，用于分析自然数的分区：Omega运算符演算。更准确地说，我们构造一个对绝对会聚矩阵值展开进行线性运算的算子，该运算符选择那些展开的适当项。在我们的框架中，可以使用矩阵值函数的新表示形式。我们的表示很简单，只需要计算矩阵逆和标量函数的泰勒级数的基本运算即可。为了证明我们方法的有效性，我们获得了与ODE基本理论，扰动计算，涉及矩阵指数的多重积分，Sylvester方程，