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.     

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．     

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.     

关于极分解和广义极分解的一些新结果

本文研究极分解和广义极分解.孙和陈提出的Frobenius范数下的逼近定理被推广至任何酉不变范数情形.得到了次酉极因子的一个新的表达式.通过新的表达式,我们得到了次酉极因子在任何酉不变范数下的扰动界.最后,讨论了数值计算方法.     

关于Haar 酉元的一个注记

设M 是一个Ⅱ₁ 型因子, τ 是M 的正规的、忠实的迹态, U ∈ M 是一个Haar 酉元, p ∈ M是一个投影, τ (p) = (1/n) (n > 3, n ∈ Z), p 和U 自由. 我们用初等方法证明了若pUp = wh 是pUp 的极分解, 则w 也是一个Haar 酉元且w 和h 是自由的. 我们还给出了pUp 的矩的刻画.     

关于矩阵方程X+A*X-1A=P的解及其扰动分析

考虑非线性矩阵方程X＋A^＊（X^-1）A=P其中A是n阶非奇异复矩阵,P是n阶Hermite正定矩阵．本文给出了Hermite正定解和最大解的存在性以及获得最大解的一阶扰动界,改进了文[5,6]中的部分结论．     

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.     

四元数矩阵的酉相合与酉相似

本文讨论了四元数矩阵的酉相合,四元数矩阵的酉相合是复矩阵的复酉相合的自然推广,并且它有许多好的性质.由于四元数矩阵的酉相合与酉相似有着密切联系,本文还讨论了四元数矩阵的酉相似的一些性质.     

酉延拓矩阵的奇异值分解及其广义逆

从普通奇异值分解出发,导出了酉延拓矩阵的奇异值和奇异向量与母矩阵的奇异值和奇异向量间的定量关系,同时对酉延拓矩阵的满秩分解及g逆,反射g逆,最小二乘g逆,最小范数g逆作了定量分析,得到了酉延拓矩阵的满秩分解矩阵F*和G*与母矩阵A的分解矩阵F和G之间的关系.最后给出了相应的快速求解算法,并举例说明该算法大大降低了分解的计算量和存储量,提高了计算效率.     

关于酉对称矩阵的QR分解及其算法

为了简化大型行(列)酉对称矩阵的QR分解,研究了行(列)酉对称矩阵的性质,获得了一些新的结果,给出了行(列)酉对称矩阵的QR分解的公式和快速算法,它们可极大地减少行(列)酉对称矩阵的QR分解的计算量与存储量,并且不会丧失数值精度.同时推广和丰富了邹红星等(2002)的研究内容,拓宽了实际应用领域的范围.     

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.     

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.     

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.     

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.     

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.     

A combinatorial characterisation of embedded polar spaces

Some classical polar spaces admit polar spaces of the same rank as embedded polar spaces (often arisen as the intersection of the polar space with a non-tangent hyperplane). In this article we look at sets of generators that behave combinatorially as the set of generators of such an embedded polar space, and we prove that they are the set of generators of an embedded polar space.     

Partial Ovoids in Classical Finite Polar Spaces

Ovoids in finite polar spaces are related to many other objects in finite geometries. In this article, we prove some new upper bounds for the size of partial ovoids in Q ^–(2n+1,q) and W(2n+ 1,q). Further, we give a combinatorial proof for the non-existence of ovoids of H(2n +1,q ²) for n>q ³.     

Sub-weakly Embedded Singular and Degenerate Polar Spaces

We show that every sub-weak embedding of any singular (degenerate or not) orthogonal or unitary polar space of non-singular rank at least 3 in a projective space PG , a commutative field, is the projection of a full embedding in some subspace PG of PG , where PG contains PG and is a subfield of . The same result is proved in the symplectic case under the assumption that the field over which the polarity is defined is perfect if the characteristic is 2 and if each secant line of the embedded polar space contains exactly two points of . This completes the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces (singular or not; degenerate or not) of non-singular rank at least 3 and defined over a commutative field , where in the characteristic 2 case is perfect if the polar space is symplectic and the degree of the embedding is 2.