The least squares g-inverses for sum of matrices

Abstract

In this article, we exhibit under suitable conditions a neat relationship between the least squares g-inverse for a sum of two matrices and the least squares g-inverses of the individual terms. We give a necessary and sufficient condition for the set equations (A + B){1, 3} = A{1, 3} + B{1, 3} and (A + B){1, 4} = A{1, 4} + B{1, 4}.

Keywords:

• the least squares g-inverse
• the minimum norm g-inverse
• maximal rank
• minimal rank
• generalized parallel sum

AMS Subject Classifications::

• 15A03
• 15A09
• 65F05

Acknowledgements

The authors would like to thank Prof. Ravindra Bapat and the anonymous referees for their very detailed comments, which greatly improved the presentation of this article. The work was supported by the start-up fund of Wuyi University, Jiangmen 529020 and the National Natural Science Foundation of China (Grant No. 11171371).