Characterization of matrices satisfying the reverse order law for the Moore–Penrose pseudoinverse
Oskar Kędzierski · Linear and Multilinear Algebra · 2025
For any complex matrix A, there exists a unique complex matrix A†, called the Moore–Penrose pseudoinverse, such that the following conditions, known as the Penrose conditions, hold: AA†A=A, A†AA†=A†, AA† is Hermitian and A†A is Hermitian. However, the condition (AB)†=B†A†, known as the reverse-order law, does not hold in general. We provide a new constructive characterization of matrices that satisfy the reverse-order law. In particular, for a given matrix A, we construct another matrix B, of arbitrary compatible size and rank, in terms of the singular value decomposition of matrix A. Moreover, we show that any matrix B satisfying the reverse-order law for a fixed A arises from a similar construction. As a consequence, we show that B†A† is the Moore–Penrose pseudoinverse of AB if and only if (BB∗)†(A∗A)† is the Moore–Penrose pseudoinverse of A∗ABB∗. In addition, we prove similar equivalent characterizations and conditions for B†A† being a {1,2}-, {1,2,3}-, or {1,2,4}-inverse of AB, that is, a matrix that satisfies only some of the four Penrose conditions. These characterizations provide geometric insight in terms of the principal angles between the column spaces of A∗ and B.