SOME GENERALIZED FORMS OF LEAST-SQUARES G-INVERSE, MINIMUM NORM G-INVERSE, AND MOORE-PENROSE INVERSE MATRICES

For given matrices A, M and L of orders n × m, n × q and m × r respectively, we consider various types of g-inverses of A by imposing some rank conditions, i.e., rank(A′M) = rank(AL) = rank(A), on these matrices. Further, we introduce two square matrices H = A(M′A)- and F = L(AL)-A, which turn out to be idempotent under the rank conditions. In the light of these properties, we develop some equivalent conditions on g-inverses of A. Based on these results, the L-inverse, M-inverse and LMN inverse (when N = 0) developed by Rao and Yanai (1985a) as generalized forms of least-squares g-inverse, minimum norm g-inverse and Moore-Penrose g-inverse matrices, respectively, are shown to be special cases of the g-inverse matrices which we introduce in section 3 without assuming the nonnegative definite condition. © 1990.