Miscellaneous reverse order laws for generalized inverses of matrix products 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, which are natural extensions of reverse order laws for the standard inverses of products of nonsingular matrices of the same size. Under the assumption that A, B, and C are singular matrices of the appropriate sizes, two reverse order laws for generalized inverses of the matrix products AB and ABC can be written as $$(AB)^{(i,\ldots ,j)} = B^{(s_2,\ldots ,t_2)}A^{(s_1,\ldots ,t_1)}$$ and $$(ABC)^{(i,\ldots ,j)} = C^{(s_3,\ldots ,t_3)}B^{(s_2,\ldots ,t_2)}A^{(s_1,\ldots ,t_1)}$$ , respectively, 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 laws to hold because there are all 15 types of $$\{i,\ldots , j\}$$ -generalized inverse for a given matrix according to the combinatoric choice of the four Penrose equations. In this paper, we first establish four groups of of mixed reverse order laws for $$\{1\}$$ - and $$\{1,2\}$$ -generalized inverses of AB and ABC. We then give a classified investigation to a family of reverse order laws $$(ABC)^{(i,\ldots ,j)} = C^{-1}B^{(s,\ldots ,t)}A^{-1}$$ for the eight commonly-used types of generalized inverses using the definitions of generalized inverses, the block matrix methodology, and the matrix rank methodology. A variety of consequences and applications of these reverse order laws are presented, and a list of open problems on reverse order laws are mentioned.