Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse
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 $\mathcal{C}(AAB)=\mathcal{C}(BB*A^)$ 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 $\mathcal{C}(B)$.
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