Further results on the reverse-order law

AbstractAn explicit expression is obtained for a pair of generalized inverses (B−,A−) such that B−A−=(AB)+MN, and a class of pairs (B−,A− of this property is shown. A necessary and sufficient condition for (AB)− to have the expression B−A− is also given

Reverse Order Law for the Core Inverse in Rings

In this paper, necessary and sufficient conditions of the onesided reverse order law (ab)((sic)) = b((sic))a((sic)) , the two-sided reverse order law (ab)((sic)) = b((sic))a((sic)) and (ba)((sic)) = a((sic))b((sic)) for the core inverse are given in rings with involution. In addition, the mixed-type reverse order laws, such as (ab)((sic)) = b((sic))(abb((sic)))((sic)) , a((sic)) = b(ab)((sic)) and (ab)((sic)) = b((sic)) a((sic)) , are also considered.- This research was supported by China Postdoctoral Science Foundation (No. 2018M632385), the National Natural Science Foundation of China (No. 11771076), the Natural Science Foundation of Jiangsu Province (No. BK20141327), the Portuguese Funds through FCT-"Fundacao para a Ciencia e a Tecnologia", within the project UID/MAT/00013/2013

Further results on the reverse order law for the group inverse in rings

In this paper, we use the Drazin inverse to derive some new equivalences of the reverse order law for the group inverse in unitary rings. Moreover, if the ring has an involution, we present more equivalences when both involved elements are EP. 2013 Elsevier Inc. All rights reserved.The first and the second authors have been supported by the Guangxi Natural Science Foundation (013GXNSFAA019008), the Key Project of Education Department of Guangxi Project (201202ZD031), and by National Science Foundation of China (11361009, 11061005). The third author has been supported by the Universidad Politecnica de Valencia (PAID-06-12).Liu, X.; Zhang, M.; Benítez López, J. (2014). Further results on the reverse order law for the group inverse in rings. Applied Mathematics and Computation. 229:316-326. https://doi.org/10.1016/j.amc.2013.12.030S31632622

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