On reverse-order laws for least-squares g-inverses and minimum norm g-inverses of a matrix product

Summary.

Let A^(1,3) and A^(1,4) denote a least-squares g-inverse and a minimum norm g-inverse of a matrix A, respectively. In this paper, we establish necessary and sufficient conditions for ${\{B^{(1,3)}A^{(1,3)}\} \subseteq \{(AB)^{(1,3)}\}}$ and ${\{B^{(1,4)}A^{(1,4)}\} \subseteq \{(AB)^{(1,4)}\}}$ to hold. We also show that the well-known reverse-order law ${(AB)^{\dagger}= B^{\dagger} A^{\dagger}}$ is equivalent to ${\{B^{(1,3)}A^{(1,3)}\} \subseteq \{(AB)^{(1,3)}\}}$ and ${\{B^{(1,4)}A^{(1,4)}\} \subseteq \{(AB)^{(1,4)}\}}$ .