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

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.