Rank Equalities Related to Generalized Inverses of Matrices and Their Applications

This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions that involve variant matrices, the fundamental work is concerning extreme ranks of the two linear matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications, we present a wide range of their consequences and applications in matrix theory.