# Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse

Source: [Characterization of Matrices Satisfying the Reverse Order Law for the Moore-Penrose Pseudoinverse](<https://www.sota2.com/research/paper/68fa5bac1e209131ce9ca82e>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T17:12:40.380Z

Authors: Oskar K\\k{e}dzierski

Published: 2024-04-03T16:22:32.000Z

## 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 $\\mathcal{C}(B)$.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2404.02843>)

[PDF](<https://arxiv.org/pdf/2404.02843.pdf>)