A generalization of the group inverse

Abstract

From the point of view of generalizations, the core inverse of a matrix has received a lot of attention of researchers in this area resulting  in many generalizations of the core inverse, like core-EP, DMP, BT, WC and so on. However, the same can not be said for the group  inverse. In this paper we introduce a new g-inverse for a matrix of arbitrary index that generalizes the group inverse as it coincides with  its group inverse in case the matrix has index at most 1 and call it GG (generalized group) inverse. We study several properties and  characterizations of this extension by using the core-EP decomposition. Apart from investing it for its properties and characterizations,  we study the several features it has common with the group inverse. In addition, by using Drazin and GG inverses we introduce a new  class of matrices that extends the concept of WC matrix recently defined in the literature [4].