Home > ELA > Vol. 30 (2015)

#### Keywords

Group inverse, Moore-Penrose inverse, Sign pattern, S2GI-matrix

#### Abstract

Let $M= \left[ \begin{array}{cc} A& B \\ C& O \end{array} \right]$ be a complex square matrix where A is square. When BCB^{\Omega} =0, rank(BC) = rank(B) and the group inverse of $\left[ \begin{array}{cc} B^{\Omega} A B^{\Omega} & 0 \\ CB^{\Omega} & 0 \right]$ exists, the group inverse of M exists if and only if rank(BC + A)B^{\Omega}AB^{\Omega})^{\pi}B^{\Omega}A)= rank(B). In this case, a representation of $M^#$ in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a (0,+,−)-matrix obtained from A by replacing each entry by its sign. The qualitative class of A is the set of the matrices with the same sign pattern as A, denoted by Q(A). The matrix A is called S^2GI, if the group inverse of each matrix \bar{A} in Q(A) exists and its sign pattern is independent of e A. By using the group inverse representation, a necessary and sufficient condition for a real block matrix to be an S^2GI-matrix is given.

#### Recommended Citation

Sun, Lizhu; Wang, Wenzhe; Bu, Changjiang; Wei, Yimin; and Zheng, Baodong.
(2015),
"Representations and sign pattern of the group inverse for some block matrices",
*Electronic Journal of Linear Algebra*,
Volume 30, pp. 744-759.

DOI: https://doi.org/10.13001/1081-3810.3169

*Abstract*