ELA, Volume 13, pp. 249-261, November 2005, abstract.

Special Forms of Generalized Inverses of Row Block Matrices

Yongge Tian

Given a row block matrix [A, B], this paper investigates the relations
between the generalized inverse [A, B]^- and the column block matrix
[(A^-)^T  (B^-)^T]^T consisting of two generalized inverses A^- and B^-. 
The first step of the investigation is to establish a formula for the 
minimal rank of the difference 
             [A, B\,]^-  -  [(A^-)^T  (B^-)^T]^T, 
the second step is to find a necessary and sufficient condition for 
             [A, B\,]^-  =  [(A^-)^T  (B^-)^T]^T 
to hold by letting the minimal rank be zero. Seven types of generalized 
inverses of matrices are taken into account.