On mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product

If A and B are a pair of invertible matrices of the same size, then the product AB is nonsingular, too, and the inverse of the product AB satisfies the reverse-order law (AB)−1 = B−1A−1. This law can be used to find the properties of (AB)−1, as well as to simplify various matrix expressions that involve the inverse of a matrix product. However, this formula cannot trivially be extended to the Moore-Penrose inverse of matrix products. For a generalm×n complex matrix A, the Moore-Penrose inverse A† of A is the unique n×m matrix X that satisfies the following four Penrose equations: (i) AXA=A, (ii) XAX =X, (iii) (AX)∗ =AX, (iv) (XA)∗ =XA, where (·)∗ denotes the conjugate transpose of a complex matrix. A matrix X is called a {1}-inverse (inner inverse) ofA if it satisfies (i) and is denoted byA−. General properties of the Moore-Penrose inverse can be found in [2, 4, 16]. LetA and B be a pair of matrices such thatAB exists. In many situations, one needs to find the Moore-Penrose inverse of the product AB and its properties. Because A†A, BB†, and BB†A†A are not necessarily identity matrices, the relationship between (AB)† and B†A† is quite complicated and the reverse-order law (AB)† = B†A† does not necessarily hold. Therefore, it is not easy to simplify matrix expressions that involve the MoorePenrose inverse of matrix products. Theoretically speaking, for any matrix product AB, the Moore-Penrose inverse (AB)† can be written as (AB)† = B†A† or (AB)† = B†A†+X, (1)