Classification analysis to the equalities A(i,…,j) = B(k,…,l) for generalized inverses of two matrices
Yongge Tian · Linear and Multilinear Algebra · 2019
One of the fundamental research problems in the theory of generalized inverses has certainly been establishments of various matrix equalities that involve generalized inverses. A simplest form of such matrix equalities is given by A(i,…,j)=B(k,…,l), where A and B are matrices of the same size, and (⋅)(i,…,j) denotes the {i,…,j}-generalized inverse of a matrix. Because generalized inverses of a matrix are not necessarily unique, the equality A(i,…,j)=B(k,…,l) does not imply A=B for singular matrices. In this paper, we derive necessary and sufficient conditions for this kind of equalities to hold for the eight commonly-used types of generalized inverse of A and B using the matrix equation method and the matrix rank method.