Reverse order law for reflexive generalized inverses of products of matrices

Abstract

The relationships between generalized inverses of the product of two matrices A. B and the product of generalized inverses of A, B have been studied in the literature. By applying the product singular value decomposition (P-SVD), in this paper we derive equivalent conditions for B{1, 2} A{1, 2} ⊆ ( AB){1, 2}, ( AB){1, 2} ⊆ B{1, 2} A{1, 2}, B{1, 2} A{1, 2} = ( AB){1, 2}, and equivalent conditions for ( AB){1} ⊆ B{1} A{1}.