Generalized inverses and a block-rank equation

摘要

For a square matrix A of index 1, the block-rank equationrankABCX=rank(A)is studied. Geometrical conditions are given to characterize the solution of this equation. Further, all matrices B and C are described for the solution X=A#, where A# is the group inverse of A. In addition, we extend these results to reflexive generalized inverses. This contributes to a result recently obtained by Wei [SIAM J. Matrix Anal. Appl. 17 (1996) 744] and it is a generalization of a result by Groß [Lin. Alg. Appl. 289 (1999) 127].