Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse

We give a constructive characterization of matrices satisfying the reverse-order law for the Moore–Penrose pseudoinverse. In particular, for a given matrix $A$ we construct another matrix $B$, of arbitrary compatible size and chosen rank, in terms of the right singular vectors of $A$, such that the reverse order law for $AB$ is satisfied. Moreover, we show that any matrix satisfying this law comes from a similar construction. As a consequence, several equivalent conditions to $B^{+}{A}^{+}$ being a pseudoinverse of $AB$ are given, for example $\mathcal{C}(A^{\ast }AB)=\mathcal{C}(B{B}^{\ast }{A}^{\ast })$ or $B{\left(AB\right)}^{+}A$ being an orthogonal projection. In addition, we parameterize all possible SVD decompositions of a fixed matrix and give Greville-like equivalent conditions for $B^{+}{A}^{+}$ being a $\{1,2\}$-,$\{1,2,3\}$- and $\{1,2,4\}$-inverse of $AB$, with a geometric insight in terms of the principal angles between $\mathcal{C}(A^{\ast })$ and $\mathcal{C}(B)$.