Miscellaneous reverse order laws and their equivalent facts for generalized inverses of a triple matrix product

Abstract

Reverse order laws for generalized inverses of products of matrices are a class of algebraic matrix equalities that are composed of matrices and their generalized inverses, which can be used to describe the links between products of matrix and their generalized inverses and have been widely used to deal with various computational and applied problems in matrix analysis and applications. ROLs have been proposed and studied since 1950s and have thrown up many interesting but challenging problems concerning the establishment and characterization of various algebraic equalities in the theory of generalized inverses of matrices and the setting of non-commutative algebras. The aim of this paper is to provide a family of carefully thought-out research problems regarding reverse order laws for generalized inverses of a triple matrix product ABC of appropriate sizes, including the preparation of lots of useful formulas and facts on generalized inverses of matrices, presentation of known groups of results concerning nested reverse order laws for generalized inverses of the product AB, and the derivation of several groups of equivalent facts regarding various nested reverse order laws and matrix equalities. The main results of the paper and their proofs are established by means of the matrix rank method, the matrix range method, and the block matrix method, so that they are easy to understand within the scope of traditional matrix algebra and can be taken as prototypes of various complicated reverse order laws for generalized inverses of products of multiple matrices.

Keywords

• matrix product
• moore–penrose generalized inverse
• reverse order law
• orthogonal projector
• rank
• range