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 $\{1,2\}-$inverse of $AB$, with a geometric insight in terms of the principal angles between $\mathcal{C}(A^*)$ and $\mathcal{C}(B)$.

翻译：本文给出了满足Moore-Penrose伪逆反转律的矩阵的构造性刻画。具体地，对于给定矩阵$A$，我们基于其右奇异向量构造了另一任意相容尺寸且选定秩的矩阵$B$，使得$AB$的反转律成立。进一步证明，所有满足该律的矩阵均源于相似构造。由此给出了$B^+ A^+$是$AB$的伪逆的几个等价条件，例如$\mathcal{C}(A^*AB)=\mathcal{C}(BB^*A^*)$或$B\left(AB\right)^+A$为正交投影。此外，我们参数化了固定矩阵的所有可能SVD分解，并给出了$B^+A^+$为$AB$的$\{1,2\}-$逆的Greville型等价条件，从$\mathcal{C}(A^*)$与$\mathcal{C}(B)$间主角度的几何视角进行了阐释。