华东师范大学学报(自然科学版) ›› 2015, Vol. 2015 ›› Issue (1): 42-50.doi: 10.3969/j.issn.1000-5641.2015.01.005

• 应用数学与基础数学 • 上一篇    下一篇

I{2}和M{2}的有效刻画(英)

征道生   

  1. 华东师范大学数学系,  上海 200241
  • 收稿日期:2013-10-01 出版日期:2015-01-25 发布日期:2015-03-29
  • 通讯作者: 征道生，男，教授, 研究方向: 数值代数与矩阵论.. E-mail:dszheng@math.ecnu.edu.cn

Efficient characterization for I{2} and M{2}

 ZHENG  Dao-Sheng   

  • Received:2013-10-01 Online:2015-01-25 Published:2015-03-29

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摘要： Stewart 给出了一个矩阵{2}-逆集合M{2}的刻划公式。 但公式.中含有多余的任意参数. 按Ben-Israel 的说法, 它不是一个有效刻划. 利用方阵的满秩分解, 本文定理 2..1 和2.2为I{2} 的一个真子集B剔除了Stewart公式.中的多余任意参数, 得到了B的有效刻划公式;. 还证明了 I{2}是其有限个子集的并集, 其中每个子集与B等距同构. 由此可分别建立I{2}, I_{2}, M{2} 和 M{2} 的有效刻划公式. 算法2..1 则可用于无重复地计算 I{2}的每个元素. 

关键词: 广义逆矩阵类的有效刻划, 矩阵的满秩分解, 并集, 集合的等距同构;

Abstract: An important characterization formula for M{2} was given by Stewart where M 2 Cm×n. But this formula contains redundant arbitrary parameters, and therefore is nonefficient. This paper, by using the matrix full rank decomposition, showed that for a proper subset of I{2}s, which is denoted as B1, the redundant arbitrary parameters in Stewart’s formula can be eliminated, and I{2}s is a union set of its certain subsets, and each of the subsets is 2-norm isometry with B1. Finally, the efficient characterization fonmulas for I{2}s, I{2} and M{2} are obtained respectively. An algorithm was provided that can be used to compute any element of I{2}s, and avoid the repeat computation work for each
element of I{2}s.

Key words: efficient characterization for a generalized inverse matrix set, 2-norm isometry of two sets, matrix full-rank factorization, solution set of a matrix equation

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