Weighted G-Drazin inverses and a new pre-order on rectangular matrices

[EN] This paper deals with weighted G-Drazin inverses, which is a new class of matrices introduced to extend (to the rectangular case) G-Drazin inverses recently considered by Wang and Liu for square matrices. First, we define and characterize weighted G-Drazin inverses. Next, we consider a new pre-order defined on complex rectangular matrices based on weighted G-Drazin inverses. Finally, we characterize this pre-order and relate it to the minus partial order and to the weighted Drazin pre-order. (C) 2017 Elsevier Inc. All rights reserved.This paper was partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria, grant resol. no. 155/14. The first and third authors were partially supported by Ministerio de Economia y Competitividad of Spain (grant no. DGI MTM2013-43678-P) and the third author was also partially supported by Ministerio de Economia y Competitividad of Spain (Red de Excelencia MTM2015-68805-REDT).Coll, C.; Lattanzi, M.; Thome, N. (2018). Weighted G-Drazin inverses and a new pre-order on rectangular matrices. Applied Mathematics and Computation. 317:12-24. https://doi.org/10.1016/j.amc.2017.08.047S122431

The W-weighted Drazin-star matrix and its dual

[EN] After decades studying extensively two generalized inverses, namely Moore--Penrose inverse and Drazin inverse, currently, we found immersed in a new generation of generalized inverses (core inverse, DMP inverse, etc.). The main aim of this paper is to introduce and investigate a matrix related to these new generalized inverses defined for rectangular matrices. We apply our results to the solution of linear systems.The authors wish to thank the editor and reviewers sincerely for their constructive comments and suggestions that have improved the quality of the paper. This research is supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX18_0053), the China Scholarship Council (File No. 201906090122), the National Natural Science Foundation of China (No.11771076, 11871145). The third author is partially supported by Ministerio de Economía y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT) and Universitat Nacional de La Pampa, Facultad de Ingeniería (Grant Resol. No. 135/19)Zhou, M.; Chen, J.; Thome, N. (2021). The W-weighted Drazin-star matrix and its dual. The Electronic Journal of Linear Algebra. 37:72-87. https://doi.org/10.13001/ela.2021.5389S72873

On some new pre-orders defined by weighted Drazin inverses

In this paper, we investigate new binary relations defined on the set of rectangular complex matrices based on the weighted Drazin inverse and give some characterizations of them. These relations become pre-orders and improve the results found by the authors in Hernandez et al. (2013) as well as extend those known for square matrices. On the other hand, some new weighted partial orders are also defined and characterized. The advantages of these new relations compared to the ones considered in the mentioned paper are also pointed out.N. Thome was partially supported by Ministerio de Economia y Competitividad of Spain (Grant DGI MTM2013-43678-P and Red de Excelencia MTM2015-68805-REDT).Hernández, AE.; Lattanzi, MB.; Thome Coppo, NJ. (2016). On some new pre-orders defined by weighted Drazin inverses. Applied Mathematics and Computation. 282:108-116. https://doi.org/10.1016/j.amc.2016.01.055S10811628

Representations of the weighted WG inverse and a rank equation's solution

In this paper, we present several representations of the W-weighted WG inverse. These representations are expressed in terms of matrix powers as well as in terms of matrix products involving only the Moore–Penrose inverse. In addition, a new characterization of the W-weighted WG inverse is presented by using a rank equation.Fil: Ferreyra, David Eduardo. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Orquera, Valentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; ArgentinaFil: Thome Coppo, Néstor Javier. Universidad Politécnica de Valencia; Españ

Representations and properties of the W-weighted core-EP inverse

In this paper, we investigate the weighted core-EP inverse introduced by Ferreyra, Levis and Thome. Several computational representations of the weighted core-EP inverse are obtained in terms of singular-value decomposition, full-rank decomposition and QR decomposition. These representations are expressed in terms of various matrix powers as well as matrix product involving the core-EP inverse, Moore-Penrose inverse and usual matrix inverse. Finally, those representations involving only Moore-Penrose inverse are compared and analyzed via computational complexity and numerical examples.This research is supported by the National Natural Science Foundation of China (No.11771076), the Scientific Innovation Research of College Graduates in Jiangsu Province (No.KYZZ16 0112), Partially supported by FCT- ‘Fundação para a Ciência e a Tecnologia’, within the project UID-MAT-00013/2013