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

摘要

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 ${{B^{(1,4)}A^{(1,4)}} subseteq {(AB)^{(1,4)}}}$ .