Moore-Penrose pseudo-inverse, generalized inverses of a matrix product, forward order law, reverse order law
New expressions are given for the Moore-Penrose inverse of a product $AB$ of two complex matrices. Furthermore, an expression for $(AB)\dg - B\dg A\dg$ for the case where $A$ or $B$ is of full rank is provided. Necessary and sufficient conditions for the forward order law for the Moore-Penrose inverse of a product to hold are established. The perturbation results presented in this paper are applied to characterize some mixed-typed reverse order laws for the Moore-Penrose inverse, as well as the reverse order law.
Castro-Gonzalez, Nieves and Hartwig, Robert E..
"Perturbation results and the forward order law for the Moore-Penrose inverse of a product",
Electronic Journal of Linear Algebra,
Volume 34, pp. 514-525.