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 $\mathcal{C}(B)$.

翻译：本文给出了满足Moore–Penrose伪逆逆序律的矩阵的构造性表征。具体而言，对给定的矩阵$A$，我们通过其右奇异向量构造另一个矩阵$B$（具有任意兼容尺寸和选定秩），使得$AB$满足逆序律。此外，我们证明了所有满足该律的矩阵均源自类似的构造。作为推论，给出了$B^+ A^+$成为$AB$伪逆的若干等价条件，例如$\mathcal{C}(A^*AB)=\mathcal{C}(BB^*A^*)$或$B\left(AB\right)^+A$为正交投影。进一步地，我们参数化了固定矩阵的所有可能的SVD分解，并给出了$B^+A^+$成为$AB$的$\{1,2\}$-、$\{1,2,3\}$-和$\{1,2,4\}$-逆的Greville型等价条件，同时通过$\mathcal{C}(A^*)$与$\mathcal{C}(B)$之间的主角度给出了几何解释。