•  
  •  
 

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.

vol30_pp744-759.tex (53 kB)
Abstract

Share

COinS
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.