Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse
Oskar Kędzierski
TL;DR
This work addresses when the reverse-order law $(AB)^+=B^+A^+$ holds for Moore–Penrose inverses, offering a constructive characterization: for a fixed $A$, one can explicitly build a compatible $B$ from $A$'s right singular vectors, and every $B$ satisfying the law arises from a similar construction. It develops a suite of equivalent Greville-type conditions and provides a geometric interpretation in terms of principal angles between $\\mathcal{C}(A^*)$ and $\\mathcal{C}(B)$, along with a parametric SVD framework for these matrices. The authors also connect these results to projections, derive multiple equivalent conditions, and supply MATLAB code to realize the construction in practice. Overall, the paper extends known results (e.g., Schwerdtfeger, Tian, Greville) and yields concrete, verifiable criteria for designing matrix pairs that satisfy the reverse-order law, with clear implications for numerical linear algebra and pseudoinverse computations.
Abstract
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 $\mathcal{C}(B)$.