广义特征值的扰动界限(Ⅰ)

到目前为止,关于广义特征值的扰动,已经建立了一些界限估计,但一般正则对的扰动界限难以算出.首先,定义某些基本参数,并利用这些参数建立几个关于一般正则矩阵对的广义特征值的扰动定理.这些定理给出的扰动界限的上界估计,一般是可以算出的.     

广义特征值的扰动界限(Ⅰ)

到目前为止,关于广义特征值的扰动,已经建立了一些界限估计,但一般正则对的扰动界限难以算出.首先,定义某些基本参数,并利用这些参数建立几个关于一般正则矩阵对的广义特征值的扰动定理.这些定理给出的扰动界限的上界估计,一般是可以算出的.     

广义奇异值的扰动

本文给出了一对矩阵的广义奇异值扰动的一致上界,并由之可导出普通奇异值扰动的经典定理。     

广义逆矩阵特征性质的注记

本文运用线性算子方法进一步探讨了一般数域上矩阵广义逆的特征性质,同时也分别简化和修正了文献［1］中定理2.3.5和定理2.3.6给出的矩阵广义逆A-的结果.     

模糊矩阵三种广义逆的判定定理

给出模糊矩阵存在广义{1,3}逆,广义{1,4}逆和Moore-Penrose广义逆的判定定理.     

Banach空间中Drazin逆的新扰动定理

讨论Banach空间中有界线性算子的Drazin逆的扰动问题.利用Jiu Ding在2003年给出的广义Neumann引理,给出关于Drazin逆的一个新扰动定理,并给出误差估计,推广了文献中相应的扰动结果.     

闭算子的线性斜投影广义逆的新扰动

讨论Banach空间中无界闭算子的线性斜投影广义逆的稳定性问题,利用T有界及广义Neumann引理给出了新的扰动定理.     

群逆的扰动界及其在奇异线性系统中的应用

本文建立了群逆的扰动界,此界基于矩阵A的Jordan标准形和P-范数,其中P是非异矩阵满足 是非异上双对角阵且 当矩阵A和A+E有相同的秩且 较小时,得到了 较好的估计.在相同的条件下,研究了相容的奇异线性系统Aχ=b的扰动,给出了χopt=A#b扰动的上界,其中A#是A的群逆,χopt是最小P-范数解.     

布尔矩阵最大广义逆

本文给出了布尔矩阵广义逆地一个充要条件。证明了定理2中构造的 A_o 恰好是 A的最大广义逆。依据这些结果,详细地研究了最大广义逆。并发现,非奇异矩阵 A 的最大广义逆 A_o与通常逆矩阵有类似的性质,如[A_o]_o=A 等等。     

Banach空间中基于Neumann引理的拟线性广义逆扰动定理

本文给出Banach空间中闭线性算子的Moore-Penrose有界拟线性投影广义逆的一种新的扰动分析方法.运用的核心工具是广义Neumann引理,这与以往其他结果中所运用的广义Banach引理的处理方法极为不同,得到了闭线性算子的MoorePenrose有界拟线性广义逆的又一个扰动定理及三个误差界不等式.     

Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

Some refined bounds for the perturbation of the orthogonal projection and the generalized inverse

In this paper, we consider the perturbation of the orthogonal projection and the generalized inverse for an n × n matrix A and present some perturbation bounds for the orthogonal projections on the rang spaces of A and A^?, respectively. A combined bound for the orthogonal projection on the rang spaces of A and A^? is also given. The proposed bounds are sharper than the existing ones. From the combined bounds of the orthogonal projection on the rang spaces of A and A^?, we derived new perturbation bounds for the generalized inverse, which always improve the existing ones. The combined perturbation bound for the orthogonal projection and the generalized inverse is also given. Some numerical examples are given to show the advantage of the new bounds.     

Perturbation bounds for weighted polar decomposition in the weighted unitarily invariant norm

In this paper, by generalizing the ideas of the (generalized) polar decomposition to the weighted polar decomposition and the unitarily invariant norm to the weighted unitarily invariant norm, we present some perturbation bounds for the generalized positive polar factor, generalized nonnegative polar factor, and weighted unitary polar factor of the weighted polar decomposition in the weighted unitarily invariant norm. These bounds extend the corresponding recent results for the (generalized) polar decomposition. In addition, we also give the comparison between the two perturbation bounds for the generalized positive polar factor obtained from two different methods. Copyright © 2008 John Wiley & Sons, Ltd.     

关于特征值与广义特征值的Bauer-Fike型相对扰动界

本文研究特征值与广义特征值的Bauer-Fike型相对扰动界.我们给出了一些新的结果.这些界从一定的意义上改进了以往相应的结论.     

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore–Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore–Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore–Penrose metric generalized inverse of bounded linear operators in Banach spaces.     

The Algebraic Perturbation Method for Generalized Inverses

Algebraic perturbation methods were first proposed for the solution of nonsingular linear systems by R. E. Lynch and T. J. Aird [2]. Since then, the algebraic perturbation methods for generalized inverses have been discussed by many scholars [3]-[6]. In [4], a singular square matrix was perturbed algebraically to obtain a nonsingular matrix, resulting in the algebraic perturbation method for the Moore-Penrose generalized inverse. In [5], some results on the relations between nonsingular perturbations and generalized inverses of $m\times n$ matrices were obtained, which generalized the results in [4]. For the Drazin generalized inverse, the author has derived an algebraic perturbation method in [6]. In this paper, we will discuss the algebraic perturbation method for generalized inverses with prescribed range and null space, which generalizes the results in [5] and [6]. We remark that the algebraic perturbation methods for generalized inverses are quite useful. The applications can be found in [5] and [8]. In this paper, we use the same terms and notations as in [1].     

矩阵方程ATXA=D的条件数与向后扰动分析

讨论矩阵方程ATXA=D,该方程源于振动反问题和结构模型修正.本文利用Moore-Penrose广义逆的性质,给出该方程解的条件数的上、下界估计.同时,利用Schauder不动点理论给出该方程的向后扰动界,这些结果可用于该矩阵方程的数值计算.     

Inner Generalized Inverses with Prescribed Idempotents

We define and characterize inner generalized inverses with prescribed idempotents. These classes of generalized inverses are natural algebraic extension of generalized inverses of linear operators with prescribed range and kernel. We consider the reverse order rule for inner generalized inverses of elements of a ring, some perturbation bounds, and we construct an iterative method for computing a (p, q)-inner inverse in Banach algebras.     

Challenging Problems on the Perturbation of Drazin Inverse

In order to estimate error bounds on the computed Drazin inverse of a matrix, we need to establish some perturbation theory for the Drazin inverse which is analogous to that for the Moore–Penrose inverse. In this paper, we present recent results on this topic, three problems are put forward in this direction.     

On eigenvalue perturbation bounds for Hermitian block tridiagonal matrices

In this paper, we give some structured perturbation bounds for generalized saddle point matrices and Hermitian block tridiagonal matrices. Our bounds improve some existing ones. In particular, the proposed bounds reveal the sensitivity of the eigenvalues with respect to perturbations of different blocks. Numerical examples confirm the theoretical results.