On group inverses and the sharp order

AbstractIn this work the group inverse of a matrix is used to define the #-order on square matrices of index 1. The #-order is similar to the ∗-order of Drazin [2] and the minus order of Hartwig [6, 10] and Nambooripad [17]. The #-order and the ∗-order are compared and contrasted. Many conditions are given which assure the equivalence of the various partial orders studied