Group inverse for block matrices and some related sign analysis

Abstract

The sign pattern of a real matrix M is the (0, 1, −1)-matrix obtained from M by replacing each entry by its sign. Let N ∈ ℝ ^n×n be a group invertible matrix. Let Q(N) be the set of real matrices with the same sign pattern as N. For any , if is group invertible and the group inverses of N and have the same sign pattern, then N is called an S²GI matrix. In this article, we present the existence and the representations for the group inverse of some block matrices with one or two full rank sub-blocks, and give a family of block matrices which are S²GI matrices. Applying these results, we can partially determine the sign pattern of the solution of singular linear system with index one.

Keywords:

• group inverse
• Moore–Penrose inverse
• sign pattern

AMS Subject Classifications::

• 15A09
• 15B35

Acknowledgements

The authors would like to thank Prof B. Shader and two reviewers for their valuable comments and suggestions. Y. Wei is supported by the National Natural Science Foundation of China under grant 10871051, Doctoral Program of the Ministry of Education under grant 20090071110003, 973 Program Project under Grant 2010CB327900, Shanghai Education Committee and Shanghai Science & Technology Committee under grant 09DZ2272900.