Reverse order laws for generalized inverses of products of two or three matrices with applications

Abstract: One of the fundamental research problems in the theory of generalized inverses of matrices is to establish reverse order laws for generalized inverses of matrix products. Under the assumption that $A$, $B$, and $C$ are three nonsingular matrices of the same size, the products $AB$ and $ABC$ are nonsingular as well, and the inverses of $AB$ and $ABC$ admit the reverse order laws $(AB)^{-1} = B^{-1} A^{-1}$ and $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$, respectively. If some or all of $A$, $B$, and $C$ are singular, two extensions of the above reverse order laws to generalized inverses can be written as $(AB)^{(i,\ldots,j)} = B^{(i_2,\ldots,j_2)} A^{(i_1,\ldots,j_1)}$ and $(ABC)^{(i,\ldots,j)} = C^{(i_3,\ldots,j_3)} B^{(i_2,\ldots,j_2)}A^{(i_1,\ldots,j_1)}$, or other mixed reverse order laws. These equalities do not necessarily hold for different choices of generalized inverses of the matrices. Thus it is a tremendous work to classify and derive necessary and sufficient conditions for the reverse order law to hold because there are all 15 types of ${i,\ldots, j}$-generalized inverse for a given matrix according to combinatoric choices of the four Penrose equations. In this paper, we first establish several decades of mixed reverse order laws for ${1}$- and ${1,2}$-generalized inverses of $AB$ and $ABC$. We then give a classified investigation to a special family of reverse order laws $(ABC)^{(i,\ldots,j)} = C^{-1}B^{(k,\ldots,l)}A^{-1}$ for the eight commonly-used types of generalized inverses using definitions, formulas for ranges and ranks of matrices, as well as conventional operations of matrices. Furthermore, the special cases $(ABA^{-1})^{(i,\ldots,j)} = AB^{(k,\ldots,l)}A^{-1}$ are addressed and some applications are presented.