Structured Perturbations Part I: Normwise Distances
Structured Perturbations Part I: Normwise Distances
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DOI：
10.1137/s0895479802405732
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
S. Rump
中科院分区：
文献类型：
--
作者：
S. Rump
本文研究了线性系统的条件数、矩阵求逆的条件数以及到最近奇异矩阵的距离等与范数结构扰动有关的问题。调查下的结构是对称的，过对称的，反对称的，对称Toeplitz，一般Toeplitz，循环，汉克尔，和过对称汉克尔矩阵（一些结果对其他结构，如三对角和三对角Toeplitz矩阵，对称和一般，以及）。我们表明，对于一个给定的矩阵的最坏情况下的结构化条件数的所有右手边是等于非结构化的条件数。对于一个特定的右手边，我们给出了各种显式公式和估计的条件数的线性系统，特别是条件数的比率与结构和非结构扰动。此外，矩阵求逆的条件数被证明是相同的结构和非结构扰动，并证明了相同的距离最近的奇异矩阵。它遵循经典的Eckart-Young定理的推广，即，对于上述所有结构扰动，条件数的倒数等于到最近奇异矩阵的距离。
In this paper we study the condition number of linear systems, the condition number of matrix inversion, and the distance to the nearest singular matrix, all problems with respect to normwise structured perturbations. The structures under investigation are symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Hankel matrices (some results on other structures such as tridiagonal and tridiagonal Toeplitz matrices, both symmetric and general, are presented as well). We show that for a given matrix the worst case structured condition number for all right-hand sides is equal to the unstructured condition number. For a specific right-hand side we give various explicit formulas and estimations for the condition numbers for linear systems, especially for the ratio of the condition numbers with respect to structured and unstructured perturbations. Moreover, the condition number of matrix inversion is shown to be the same for structured and unstructured perturbations, and the same is proved for the distance to the nearest singular matrix. It follows a generalization of the classical Eckart--Young theorem, namely, that the reciprocal of the condition number is equal to the distance to the nearest singular matrix for all structured perturbations mentioned above.