Characterization of matrices satisfying the reverse order law for the Moore-Penrose pseudoinverse

For any complex matrix A, there exists a unique complex matrix $ {A}<^>\dag \! $ A dagger, called the Moore-Penrose pseudoinverse, such that the following conditions, known as the Penrose conditions, hold: $ A{A}<^>\dag \!A=A $ AA dagger A=A, $ {A}<^>\dag \!A{A}<^>\dag \!={A}<^>\dag \! $ A dagger AA dagger=A dagger, $ A{A}<^>\dag \! $ AA dagger is Hermitian and $ {A}<^>\dag \!A $ A dagger A is Hermitian. However, the condition $ {( AB )}<^>\dag \!={B}<^>\dag \!{A}<^>\dag \! $ (AB)dagger=B dagger A dagger, 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}<^>\dag \!{A}<^>\dag \! $ B dagger A dagger is the Moore-Penrose pseudoinverse of AB if and only if $ {( BB<^>* )}<^>\dag \!{( A<^>*A )}<^>\dag \! $ (BB & lowast;)dagger(A & lowast;A)dagger is the Moore-Penrose pseudoinverse of $ A<^>*ABB<^>* $ A & lowast;ABB & lowast;. In addition, we prove similar equivalent characterizations and conditions for $ {B}<^>\dag \!{A}<^>\dag \! $ B dagger A dagger being a $ \{1,2\} $ {1,2}-, $ \{1,2,3\} $ {1,2,3}-, or $ \{1,2,4\} $ {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<^>* $ A & lowast; and B.