论文标题
基质扰动理论的教程（使用紧凑型矩阵符号）
A Tutorial on Matrix Perturbation Theory (using compact matrix notation)
论文作者
论文摘要
矩阵和运算符的分析扰动理论是一种非常有用的数学技术。大多数对该方法的基本介绍在物理文献中具有其背景，尤其是量子力学。在本说明中，我们对这种与任何物理概念无关的方法进行了介绍，并且纯粹依赖于线性代数的概念。此演示文稿的另一个功能是矩阵符号和方法始终使用。特别是，我们将特征值和特征向量的分析扩展的每个项制定为{\ em矩阵方程}，尤其是sylvester方程。给出了此类矩阵方程解决方案解决方案的可溶性条件和明确表达式，并根据这些解决方案给出了分析扩展中每个项的表达式。这种统一的处理简化了文献中通常看到的复杂符号，尤其是为非炎症和退化病例以及更高级术语提供了相对紧凑的表达方式。
Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In this note, we give an introduction to this method that is independent of any physics notions, and relies purely on concepts from linear algebra. An additional feature of this presentation is that matrix notation and methods are used throughout. In particular, we formulate the equations for each term of the analytic expansions of eigenvalues and eigenvectors as {\em matrix equations}, namely Sylvester equations in particular. Solvability conditions and explicit expressions for solutions of such matrix equations are given, and expressions for each term in the analytic expansions are given in terms of those solutions. This unified treatment simplifies somewhat the complex notation that is commonly seen in the literature, and in particular, provides relatively compact expressions for the non-Hermitian and degenerate cases, as well as for higher order terms.
