Several representations of generalized inverse and their application

Abstract

In this paper we first introduce a full-rank decomposition representation of the generalized inverse of a given complex matrix A, which is based on an arbitrary full-rank decomposition of G, where G is a matrix such that R(G)=T and N(G)=S. Using this representation, we obtain two maximum rank minor representations of the generalized inverse . As an application we give the Cramer's rule of the general restricted linear system. Finally, a numerical example shows that these representations are correct.

Keywords:

• generalized inverse A_T,S ⁽²⁾
• full-rank decomposition representation
• maximum rank minors
• minors representation

2000 AMS Subject Classification :

• 15A09
• 47A10

Acknowledgements

The authors wish to express their sincere thanks to the two anonymous referees for their helpful comments and suggestions. This project was granted financial support from Shanghai Science and Technology Committee (No. 062112065), China Postdoctoral Science Foundation (No. 20060400634) and The University Young Teacher Sciences Foundation of Anhui Province (No. 2006jq1220zd).