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 C(A^*AB)=C(BB^*A^*) or B(AB)⁺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\-inverse of $AB$, with a geometric insight in terms of the principal angles between C(A^*) and C(B).