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 $\mathcal{ C}(A^*AB)=\mathcal{ C}(BB^*A^*)$ 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 $\mathcal{ C}(A^*)$ and $\mathcal{ C}(B)$.
Categories: math.NA cs.NA