Weak core and central weak core inverse in a proper ∗*-ring

In this paper, we introduce the notion of weak core and central weak core inverse in a proper $*$-ring. We further elaborate on these two classes by producing a few representation and characterization of the weak core and central weak core invertible elements. We investigate additive properties and a few explicit expressions for these two classes of inverses through other generalized inverses. In addition to these, numerical examples are provided to validate a few claims on weak core and central weak core inverses.Comment: 20 pages, 1 figur

Characterization and Representation of Weighted Core Inverse of Matrices

In this paper, we introduce new representation and characterization of the weighted core inverse of matrices. Several properties of these inverses and their interconnections with other generalized inverses are explored. Through one-sided core and dual-core inverse, the existence of a generalized weighted Moore-Penrose inverse of matrices is proposed. Further, by applying a new representation and using the properties of the weighted core inverse of a matrix, we discuss a few new results related to the reverse order law for these inverses.Comment: 18 page

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-