On The Generalized Inverse For The Sum Of Multiple Matrices

The generalized inverse for a sum of matrices has quite important applica-tions 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 repre-sent of sum of matrices under certain conditions.In this thesis, for k nonzero m x n complex matrices. Firstly, we present the necessary and sufficient conditions for the set identities (A+A2+…+Aκ｛1,3｝=A1｛1,3｝+A2｛1,3｝+…+Aκ｛1,3｝and(A1+a2+…Aκ)｛1,4｝=A1｛1,4｝+A｛1,4｝+…+Aκ｛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+…+Aκ)｛1,2,,3｝=A1｛1,3｝+A2｛1,3｝+…+Aκ｛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.