Group inverse for block matrices and some related sign analysis
Jiang Zhou, Changjiang Bu, Yimin Wei · Linear and Multilinear Algebra · 2011
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.