SIAM Review, Volume 62, Issue 2, Page 463-482, January 2020.
We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is not so well known outside a community of specialists. We give two different proofs of the main eigenvector perturbation theorem. The first, a block-diagonalization technique inspired by the numerical linear algebra research community and based on the implicit function theorem, has apparently not appeared in the literature in this form. The second, based on complex function theory and on eigenprojectors, as is standard in analytic perturbation theory, is a simplified version of well-known results in the literature. The second derivation uses a convenient normalization of the right and left eigenvectors defined in terms of the associated eigenprojector, but although this dates back to the 1950s, it is rarely discussed in the literature. We then show how the eigenvector perturbation theory is easily extended to handle other normalizations that are often used in practice. We also explain how to verify the perturbation results computationally. We conclude with some remarks about difficulties introduced by multiple eigenvalues and give references to work on perturbation of invariant subspaces corresponding to multiple or clustered eigenvalues. Throughout the paper we give extensive bibliographic commentary and references for further reading.

中文翻译：

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特征值和特征向量的一阶摄动理论

SIAM评论，第62卷，第2期，第463-482页，2020年1月。
我们提出了一个简单特征值的一阶扰动分析以及一般正方形矩阵的相应右，左特征向量，而不是假定为Hermitian或法线。特征值结果是广泛的科学界所熟知的。特征向量的处理更为复杂，其扰动理论在专家社区之外并不为人所知。我们给出了主要特征向量摄动定理的两种不同证明。第一种是受数值线性代数研究界启发并基于隐函数定理的块对角化技术，显然没有以这种形式出现在文献中。第二种方法是复杂函数理论和本征投影仪的基础，这是解析扰动理论中的标准方法，是文献中著名结果的简化版本。第二种推导使用了根据相关的本征投影仪定义的左右本征向量的便捷归一化方法，但是尽管这可以追溯到1950年代，但在文献中很少讨论。然后，我们展示了特征向量扰动理论如何轻松扩展以处理实践中经常使用的其他归一化。我们还将说明如何通过计算验证扰动结果。我们以关于多个特征值引入的困难的一些结论作为结论，并为从事与多个特征值或聚类特征值相对应的不变子空间的摄动提供了参考。在整篇文章中，我们给出了大量的书目评论和参考，以供进一步阅读。