| Abstract | 多个矩阵和的广义逆在数值线性代数及其他应用领域里有非常重要的应用. 许多学者已经对此作了研究，但大多注重在在特定条件下两个矩阵和的广义逆的表示上.
本文中，对于k个非零 m×n 复矩阵Ai。首先，我们给出了集合等式(A1+A2+…Ak){1,3}=A1{1,3}+A2{1,3}+…+Ak{1,3}和(A1+A2+…+ Ak){1,4}=A1{1,4}+A2{1,4}+…+Ak{1,4}成立的充分必要条件. 其次，根据加权广义逆与传统广义逆之间的联系，我们不加证明地给出了上述关系在加权的情况下成立的充要条件. 最后，对于{1,2,3}-逆给出了集合等式(A1+A2+…Ak){1,2,3}=A1{1,3}+A2{1,3}+…+Ak{1,3}以及(A+B){1,2,3}⊇A{1,2,3}+B{1,2,3} 成立的充分必要条件.{1,2,4}-逆的情形类似. |
| Other Abstract | The generalized inverse for a sum of matrices has quite important applications in Theory of Matrix, Statistics, Theory of Control and other applied fields.
Many researchers have studied this subject, but most of them focus on the represent of sum of matrices under certain conditions.
In this thesis, for k nonzero m×n complex matrices Ai. Firstly, we present the necessary and sufficient conditions for the set identities (A1+A2+…Ak){1,3}=A1{1,3}+A2{1,3}+…+Ak{1,3} and (A1+A2+…+Ak){1,4}=A1{1,4}+A2{1,4}+…+Ak{1,4}. Secondly, according the connection between the weighted generalized inverse and the traditional generalized inverse, we show the iff conditions of the relations mentioned before. Lastly, for {1,2,3}-inverse and
{1,2,4}-inverse, we give the sufficient and necessary conditions for the following
relationships:(A1+A2+…Ak){1,2,3}=A1{1,3}+A2{1,3}+…+Ak{1,3} and (A+B){1,2,3}⊇A{1,2,3}+B{1,2,3}. The case of {1,2,4}-inverse is similar. |
