Classification analysis to the equalities A(i,...,j) = B(k,...,l) for generalized inverses of two matrices.

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. [ABSTRACT FROM AUTHOR]