On generalized-Drazin inverses and GD-star matrices

TL;DR

This paper introduces new classes of matrices called GD-star and GD-star-one, explores their properties, and establishes their representations and order relations, advancing the understanding of generalized-Drazin inverses.

Contribution

It defines and characterizes GD-star and GD-star-one matrices, providing new insights and decompositions related to generalized-Drazin inverses.

Findings

Representation of GD-star matrices via core-nilpotent and Hartwig-Spindelbock decompositions

Introduction of GD-star order and properties

Results on reverse-order and forward-order laws for GD inverse

Abstract

Motivated by the works of Wang and Liu [Linear Algebra Appl., 488 (2016) 235-248; MR3419784] and Mosic [Results Math., 75(2) (2020) 1-21; MR4079761], we provide further results on GD inverses and introduce two new classes for square matrices called GD-star (generalized-Drazin-star) and GD-star-one (generalized-Drazin-star-one) using a GD inverse of a matrix. We then exploit their various properties and characterize them in terms of various generalized inverses. We establish a representation of a GD-star matrix by using the core-nilpotent decomposition and Hartwig-Spindelbock decomposition. We also define a binary relation called GD-star order using this class of matrices. Further, we obtain some analogous results for the class of star-GD matrices. Moreover, the reverse-order law and forward-order law for GD inverse along with its monotonicity criteria are obtained.