SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 108-133, January 2021.
This paper presents a unified scheme for perturbation analysis of the Schur decomposition $A = UTU^{{H}}$ of an $n$th order matrix $A$ which allows one to obtain new local (asymptotic) componentwise perturbation bounds for the corresponding unitary transformation matrix $U$ and upper triangular matrix $T$. This scheme involves $n(n-1)/2$ basic perturbation parameters which determine all componentwise bounds for the elements of $U$, the eigenvalues, the invariant subspaces, and the superdiagonal elements of $T$. New sensitivity estimates and condition numbers of eigenvalues, invariant subspaces, and superdiagonal elements are derived which produce theoretically the same results but are computationally alternative to the well-known local perturbation bounds. These estimates are based on computing the inverse of a block lower triangular matrix of order $n(n-1)/2$, which is obtained from the Schur form, and do not involve eigenvectors. Since the computation of the inverse may be done efficiently by parallel algorithms, the implementation of new estimates can be advantageous in comparison with the usage of classical estimates.

中文翻译：

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矩阵 Schur 分解的分量微扰分析

SIAM 矩阵分析与应用杂志，第 42 卷，第 1 期，第 108-133 页，2021 年 1 月。
本文提出了一种统一的方案，用于对第 n$ 阶矩阵 $A$ 的 Schur 分解 $A = UTU^{{H}}$ 进行扰动分析，该方案允许获得新的局部（渐近）分量扰动边界对应酉变换矩阵 $U$ 和上三角矩阵 $T$。该方案涉及 $n(n-1)/2$ 基本扰动参数，这些参数确定 $U$ 的元素、特征值、不变子空间和 $T$ 的超对角元素的所有分量边界。导出了新的灵敏度估计和特征值、不变子空间和超对角元素的条件数，它们在理论上产生相同的结果，但在计算上可以替代众所周知的局部扰动边界。这些估计基于计算 $n(n-1)/2$ 阶块下三角矩阵的逆矩阵，该矩阵从 Schur 形式获得，并且不涉及特征向量。由于逆的计算可以通过并行算法有效地完成，与经典估计的使用相比，新估计的实现可以是有利的。