Solving an Open Problem About the G-Drazin Partial Order

Abstract

G-Drazin inverses and the G-Drazin partial order for square matrices have been both recently introduced by Wang and Liu. They proved the following implication: if A is below B under the G-Drazin partial order then any G-Drazin inverse of B is also a G-Drazin inverse of A. However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. In this paper, we solve completely this problem. We show that the converse, in general, is false and we provide a form to construct counterexamples. We also prove that the converse holds under an additional condition (which is also necessary) as well as for some special cases of matrices.