Nilpotent matrices and the minus partial order

[EN] In this paper, {1}-inverses of a nilpotent matrix as well as matrices above a given nilpotent matrix under the minus partial order are characterized.This paper was partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. No 155/14). The third author was partially supported by Ministerio de Economia y Competitividad of Spain (Grant number DGI MTM2013-43678-P and Grant Red de Excelencia MTM2015-68805-REDT).Gareis, MI.; Lattanzi, M.; Thome, N. (2017). Nilpotent matrices and the minus partial order. Quaestiones Mathematicae. 40(4):519-525. https://doi.org/10.2989/16073606.2017.1300612S51952540

New matrix partial order based spectrally orthogonal matrix decomposition

[EN] We investigate partial orders on the set of complex square matrices and introduce a new order relation based on spectrally orthogonal matrix decompositions. We also establish the relation of this concept with the known orders.The research of the first author was supported by the Grants [grant number RFBR-15-01-01132], [grant number MD-962.2014.1]. The second and third authors have been partially supported by Ministerio de Economia y Competitividad from Spain, DGI [grant number MTM2013-43678-P].Guterman, A.; Herrero Debón, A.; Thome, N. (2016). New matrix partial order based spectrally orthogonal matrix decomposition. Linear and Multilinear Algebra. 64(3):362-374. https://doi.org/10.1080/03081087.2015.1041365S362374643Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. doi:10.1137/1.9780898719512Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Baksalary, O. M., & Trenkler, G. (2010). Core inverse of matrices. Linear and Multilinear Algebra, 58(6), 681-697. doi:10.1080/03081080902778222Baksalary, O. M., & Trenkler, G. (2014). On a generalized core inverse. Applied Mathematics and Computation, 236, 450-457. doi:10.1016/j.amc.2014.03.048Hernández, A., Lattanzi, M., Thome, N., & Urquiza, F. (2012). The star partial order and the eigenprojection at 0 on EP matrices. Applied Mathematics and Computation, 218(21), 10669-10678. doi:10.1016/j.amc.2012.04.034Hernández, A., Lattanzi, M., & Thome, N. (2013). On a partial order defined by the weighted Moore–Penrose inverse. Applied Mathematics and Computation, 219(14), 7310-7318. doi:10.1016/j.amc.2013.02.010Hernández, A., Lattanzi, M., & Thome, N. (2015). Weighted binary relations involving the Drazin inverse. Applied Mathematics and Computation, 253, 215-223. doi:10.1016/j.amc.2014.12.102Lebtahi, L., Patrício, P., & Thome, N. (2013). The diamond partial order in rings. Linear and Multilinear Algebra, 62(3), 386-395. doi:10.1080/03081087.2013.779272Malik, S. B., Rueda, L., & Thome, N. (2013). Further properties on the core partial order and other matrix partial orders. Linear and Multilinear Algebra, 62(12), 1629-1648. doi:10.1080/03081087.2013.839676Rakić, D. S., & Djordjević, D. S. (2012). Space pre-order and minus partial order for operators on Banach spaces. Aequationes mathematicae, 85(3), 429-448. doi:10.1007/s00010-012-0133-2Nambooripad, K. S. S. (1980). The natural partial order on a regular semigroup. Proceedings of the Edinburgh Mathematical Society, 23(3), 249-260. doi:10.1017/s0013091500003801Mitra, S. K. (1987). On group inverses and the sharp order. Linear Algebra and its Applications, 92, 17-37. doi:10.1016/0024-3795(87)90248-

Left and right generalized Drazin invertible operators on Banach spaces and applications

[EN] In this paper, left and right generalized Drazin invertible operators on Banach spaces are defined and characterized by means of the generalized Kato decomposition. Then, new binary relations associated with these operators are presented and studied. In addition, a new characterization of the generalized Drazin pre-order and a sufficient condition for that to be a partial order are given by using a matrix operator technique.This paper was partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (grant Resol. No 155/14), Universidad Nacional de Rio Cuarto (grant PPI 18/C472) and CONICET (grant PIP 112-201501-00433CO). Fourth author was partially supported by Ministerio de Economia y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT).Ferreyra, DE.; Latanzi, M.; Levis, F.; Thome, N. (2019). Left and right generalized Drazin invertible operators on Banach spaces and applications. Operators and Matrices. 13(3):569-583. https://doi.org/10.7153/oam-2019-13-43S56958313

Weighted binary relations involving the Drazin inverse

[EN] The Drazin inverse of a matrix has been used in the literature to define a pre-order on the set of square complex matrices. In this paper we analyze new binary relations defined on the set of rectangular complex matrices and some relationships to the W-support idempotent. We introduce the class of weighted Drazin equal projectors and analyze the pre-orders on this class. Moreover, adjacent matrices are studied under the considered relations. Finally, some observations on weighted partial orders are given. 2014 Elsevier Inc. All rights reserved.This paper was partially supported by Universidad Nacional de LaPampa, Facultad de Ingenieria of Argentina (Grant Resol. No 049/11) and the third author was partially supported by Ministerio de Economia y Competitividad of Spain (Grant DGI MTM2013-43678P).Hernández, AE.; Lattanzi, MB.; Thome, N. (2015). Weighted binary relations involving the Drazin inverse. Applied Mathematics and Computation. 253:215-223. doi:10.1016/j.amc.2014.12.102S21522325