次酉极因子在酉不变范数下的相对扰动界

设A是一个m×n阶复矩阵,分解A=QH称为广义极分解,如果Q是m×n次酉极因子且H为n×n半正定的Hermite矩阵．本文获得了次酉极因子在任意酉不变范数下的几个相对扰动界,在某种意义上,相对扰动界比R．C．Li等获得的绝对扰动界要好．     

酉不变范数下极分解的扰动界

设A是m×n(m≥n)且秩为n的复矩阵.存在m×n矩阵Q满足Q*Q=I和n×n正定矩阵H使得A=QH,此分解称为A的极分解.本文给出了在任意酉不变范数下正定极因子H的扰动界,改进文[1,11]的结果;另外也首次提供了乘法扰动下酉极因子Q在任意酉不变范数下的扰动界.     

广义极分解

本文使用下列符号:C~(m×n)表示m×n复矩阵的集合,C_r~(m×n)表示秩为r的m×n复矩阵的集合,A~H和A~+分别表示矩阵A的共轭转置和Moore-Penrose广义逆,|| ||_2表示向量的Euclid范数和矩阵的谱范数,|| ||_F表示Frobenius范数,R(A)表示A的列     

酉对称矩阵的极分解

为了简化大型行(列)酉对称矩阵的极分解,研究了酉对称矩阵的性质,获得了一些新的结果,给出了酉对称矩阵的极分解和广义逆的公式,它们可极大地减少行(列)酉对称矩阵的极分解的计算量与存储量,并且不会丧失数值精度.同时对酉对称矩阵的极分解作了扰动分析.     

关于近似广义极因子的误差界

设A是m×n阶复矩阵,分解式A=QH称为A的广义极分解,如果Q是m×n阶次酉短阵和H是n×n半正定的Hermite矩阵．本文给出了广义极分解的一些性质和推广了有关近似极因子的相关结论．     

用割线法求矩阵的极分解

本文提出了求非奇异矩阵酉极因子的割线法,证明割线法是q-超线性收敛.并用数值例子说明割线法是有效的.     

广义非负与正极因子的乘法扰动界

本文在乘法扰动下研究了加权极分解的广义非负极因子与广义正极因子的扰动界,同时,作为特殊情形,也获得了广义极分解与极分解的非负极因子与正极因子的乘法扰动界.     

关于正交因子分解的一些结果

设G是一个图,F={F_1,F_2,…,F_d}是G的一个因子分解,H是C的一个子图,若H有d条边且恰好与每个F_i有一条公共边,则称H与F是正交的。本文研究了与图的K-因子分解正交的对集及[a,b]-子图,从而证明了关于因子分解问题的两个猜想在某些情况下成立,并提出了可进一步研究的问题。     

加权极分解

In this paper, a new matrix decomposition called the weighted polar decomposition is considered. Two uniqueness theorems of weighted polar decomposition are presented, and the best approximation property of weighted unitary polar factor and perturbation bounds for weighted polar decomposition are also studied.     

关于一些矩阵分解因子的扰动分析

关于一些矩阵分解因子的扰动分析刘新国（青岛海洋大学）ＯＮＴＨＥＰＥＲＴＵＲＢＡＴＩＯＮＢＯＵＮＤＳＦＯＲＳＯＭＥＭＡＴＲＩＸＦＡＣＴＯＲＩＺＡＴＩＯＮＳ￥ＬｉｕＸｉｎ－ｇｕｏ（ＱｃｅａｎＵｎｉｖｅｒｓｉｔｙｏｆＱｉｎｇｄａｏ，Ｑｉｎｇｄａｏ）Ａｂｓｔ...     

Some remarks on the perturbation of polar decompositions for rectangular matrices

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions for rectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms and in spectral norm, respectively, for any rectangular complex matrices, which improve recent results of Li and Sun (SIAM J. Matrix Anal. Appl. 2003; 25 :362–372). Secondly, a new absolute bound for complex matrices of full rank is given. When ‖A ? Ã‖₂ ? ‖A ? Ã‖_F, our bound for complex matrices is the same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal. Appl. 1993; 14 :588–593) for both real and complex square matrices are generalized. Copyright © 2005 John Wiley & Sons, Ltd.     

Some New Perturbation Bounds for the Generalized Polar Decomposition

The changes in the unitary polar factor under both multiplicative and additive perturbation are studied. A multiplicative perturbation bound and a new additive perturbation bound, in which a different measure of perturbation is introduced, are presented.     

稳定摄动界与厂义Cramer法则

本文分为四部分：１是引言，叙述研究这一问题的意义，特别是对鲁棒分析的意义；２是必要的基本概念与广义Ｃｒａｍｅｒ法则；３，４两部分分别研究了最小∞范数解和最小１范数解的理论和算法，本文所得到的结果不仅对线性代数，而且对鲁棒控制、最优控制和线性规划有意义．     

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.     

矩阵极分解新的数值方法

设p是大于1的偶数.本文基于方程x~p-1=0的Newton和Halley求根公式给出计算非奇异矩阵酉极因子的数值方法,并证明算法的收敛性.用数值列子说明算法的有效性.     

New perturbation bounds for the subunitary polar factors

In this article, we present some new perturbation bounds for the (subunitary) unitary polar factors of the (generalized) polar decompositions. Two numerical examples are given to show the rationality and superiority of our results, respectively. In terms of the one-to-one correspondence between the weighted case and the non-weighted case, all these bounds can be applied to the weighted polar decomposition.