Critical Points of the Singular Value Decomposition
Critical Points of the Singular Value Decomposition
复制标题
奇异值分解的关键点
DOI：
10.1137/040611719
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
K. O'Neil
中科院分区：
文献类型：
--
作者：
K. O'Neil
奇异值分解（SVD）是一个分解，在具有重复奇异值的矩阵子集上是不连续的。在本文中，在此关键集合的附近研究了SVD。显示每个单参数$ c^k $扰动横向到关键集的横向显示在临界点处唯一确定沿着扰动路径的临界点，即在扰动参数中沿optterturoTation路径（$ c^{k-1} $） 。明确发现了关键点的奇异向量的衍生物。对关键集中矩阵的扰动的奇异向量进行了申请，并将其与$ \ sin（\ theta）$定理提供的信息进行了比较。奇异向量的衍生物的估计值应用于涉及基质绝对值的不平等，例如广义的Araki-yamagami不平等。
The singular value decomposition (SVD) is a factorization that is discontinuous on the subset of matrices having repeated singular values. In this paper the SVD is studied in the vicinity of this critical set. Each one-parameter $C^k$ perturbation transversal to the critical set is shown to uniquely determine an SVD at the critical point that extends to an SVD along the perturbation path that is $C^{k-1}$ in the perturbation parameter. Derivatives of the singular vectors at the critical point are found explicitly. Application is made to the effect on the singular vectors of perturbations from a matrix in the critical set and compared to the information provided by the $\sin (\theta)$ theorem. Estimates of the derivative of the singular vectors are applied to inequalities involving the matrix absolute value, such as the generalized Araki--Yamagami inequality.