Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse
<|reference_start|>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 $\mathcal{C}(A^*AB)=\mathcal{C}(BB^*A^*)$ 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 $\mathcal{C}(B)$.<|reference_end|>
@article{kędzierski2024characterization,
    title={Characterization of Matrices Satisfying the Reverse Order Law for the
  Moore-Penrose Pseudoinverse},
    author={Oskar Kk{e}dzierski},
    journal={arXiv preprint arXiv:2404.02843},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.02843},
    primaryClass={math.NA cs.NA}
}