Using rank formulas to characterize equalities for Moore-Penrose inverses of matrix products. (Q1412627)

In order to state a matrix equality, the author finds formulas for the rank of the difference of both sizes of that equality. This technique permits to give different characterizations for the reverse order law: \((AB)^{\dagger}=B^{\dagger}A^{\dagger}\), where \(C^{\dagger} \) is the Moore-Penrose inverse of the matrix \(C\in {\mathbb C}^{m \times n}\). In a similar way, characterizations for the inverse order rule are presented for the weighted Moore-Penrose inverse: \((AB)_{MN}^{\dagger}=B_{PN}^{\dagger}A_{MP}^{\dagger}\), being \(M, N\) and \(P\) three positive definite matrices.