华东师范大学学报(自然科学版) ›› 2007, Vol. 2007 ›› Issue (1): 36-41.

• 数学 • 上一篇    下一篇

矩阵广义逆偏序的新定义

黄绚晨, 魏木生   

  1. 华东师范大学 数学系, 上海 200062
  • 收稿日期:2005-09-12 修回日期:2006-03-02 出版日期:2007-01-25 发布日期:2007-01-25
  • 通讯作者: 魏木生

New Definitions of Partial Ordering of Generalized Inverse(Chinese)

HUANG Xuan-chen, WEI Mu-sheng   

  1. Department of Mathematics, East China Normal University, Shanghai 200062, t China
  • Received:2005-09-12 Revised:2006-03-02 Online:2007-01-25 Published:2007-01-25
  • Contact: WEI Mu-sheng

摘要： 从矩阵的偏序定义出发,
提出了在集合意义下新的矩阵广义逆偏序的定义.
$\boldsymbol{A}\leqslant^{\{1\}}\boldsymbol{B}\Leftrightarrow
\boldsymbol{A}\boldsymbol{A}\{1\}=\boldsymbol{B}\boldsymbol{A}\{1\},\boldsymbol{A}\{1\}\boldsymbol{A}
=\boldsymbol{A}\{1\}\boldsymbol{B}
$以及$\boldsymbol{A}\leqslant^{\{1,2\}}\boldsymbol{B}\Leftrightarrow
\boldsymbol{A}\boldsymbol{A}\{1,2\}=\boldsymbol{B}\boldsymbol{A}\{1,2\},\boldsymbol{A}\{1,2\}
\boldsymbol{A}=\boldsymbol{A}\{1,2\}\boldsymbol{B} $.
并分别讨论了四种情况下, 矩阵$\boldsymbol{A},\boldsymbol{B}$的形式.
最后得到了相应的广义逆偏序的充要条件.

关键词: 矩阵, 广义逆, 偏序, 矩阵, 广义逆, 偏序

Abstract: By using the concept of partial ordering of matrix, some new definitions of partial ordering were put forward, such as $\boldsymbol{A}\leqslant^{\{1\}}\boldsymbol{B}\Leftrightarrow \boldsymbol{A}\boldsymbol{A}\{1\}=\boldsymbol{B}\boldsymbol{A}\{1\},\boldsymbol{A}\{1\}\boldsymbol{A}=\boldsymbol{A}\{1\}\boldsymbol{B} $ and $\boldsymbol{A}\leqslant^{\{1,2\}}\boldsymbol{B}\Leftrightarrow \boldsymbol{A}\boldsymbol{A}\{1,2\}=\boldsymbol{B}\boldsymbol{A}\{1,2\},\boldsymbol{A}\{1,2\}\boldsymbol{A}=\boldsymbol{A}\{1,2\}\boldsymbol{B}$. Four situations were discussed in detail, according to which, sufficient and necessary conditions of the new partial ordering have been derived.

Key words: generalized inverse, partial ordering, matrix, generalized inverse, partial ordering

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