Equivalent conditions for generalized inverses of products

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, and equivalent conditions for B{1}A{1} (AB){1} have been reported. By applying the product singular value decomposition, we derive more equivalent conditions for B{1}A{1} (AB){1}, and derive equivalent conditions for B{1}A{1} = (AB){1}.