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Characterizations and Representations of Core and Dual Core Inverses

Published online by Cambridge University Press:  20 November 2018

Jianlong Chen ,
Huihui Zhu ,
Pedro Patricio  and
Yulin Zhang
Jianlong Chen
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China. e-mail: jlchen@seu.edu.cn, ahzhh08@sina.com
Huihui Zhu
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China. e-mail: jlchen@seu.edu.cn, ahzhh08@sina.com CMAT-Centro de Matemática, Universidade do Minho, Braga 4710-057, Portugal
Pedro Patricio
Affiliation:
Departamento de Matemática e Aplicações, Universidade do Minho, Braga 4710-057, Portugal. e-mail: pedro@math.uminho.pt, zhang@math.uminho.pt CMAT-Centro de Matemática, Universidade do Minho, Braga 4710-057, Portugal
Yulin Zhang
Affiliation:
Departamento de Matemática e Aplicações, Universidade do Minho, Braga 4710-057, Portugal. e-mail: pedro@math.uminho.pt, zhang@math.uminho.pt CMAT-Centro de Matemática, Universidade do Minho, Braga 4710-057, Portugal
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Abstract

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In this paper, double commutativity and the reverse order law for the core inverse are considered. Then new characterizations of the Moore–Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

Keywords

15A0915A23regularitiesgroup inversesMoore-Penrose inversescore inversesdual core inversesDedekind-finite rings
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 2 , 01 June 2017 , pp. 269 - 282
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Baksalary, O. M., Problem 48-1: reverse order law for the core inverse. Image 48(2012), 40. Google Scholar
[2] Baksalary, O. M. and Trenkler, G., Core inverse of matrices. Linear Multilinear Algebra 58(2010), 681–697. http://dx.doi.org/10.1080/03081080902778222 Google Scholar
[3] Cohen, N., Herman, E. A., and Jayaraman, S., Solution to problem 48-1: reverse order law for the core inverse. Image 49(2012), 46–47. Google Scholar
[4] Drazin, M. P., Commuting properties of generalized inverses. Linear Multilinear Algebra 61(2013), no. 12, 1675–1681. http://dx.doi.org/10.1080/03081087.2012.753593 Google Scholar
[5] Hartwig, R. E., Block generalized inverses. Arch. Rational Mech. Anal. 61(1976), no. 3,197-251. http://dx.doi.org/10.1007/BF00281485 Google Scholar
[6] Patricio, P. and Mendes Araiijo, C., Moore-Penrose inverse in involutory rings: the case aa = btf. Linear Multilinear Algebra 58(2010), 445–452. http://dx.doi.org/10.1080/03081080802652279 Google Scholar
[7] Penrose, R., A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51(1955), 406–413. http://dx.doi.org/10.101 7/S0305004100030401 Google Scholar
[8] Puystjens, R. and Hartwig, R.E., The group inverse of a companion matrix. Linear Multilinear Algebra 43(1997), 137–150. http://dx.doi.org/10.1080/0308108970881 8521 Google Scholar
[9] Rakic, D. S., Dincic, N. C., and Djordjevic, D. S., Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463(2014), 115–133. http://dx.doi.org/10.1016/j.laa.2O14.09.003 Google Scholar
[10] Xu, S. Z., Chen, J. L., and Zhang, X. X., New characterizations for core inverses in rings with involution. arxiv:1512.08073v1 Google Scholar
[11] Zhu, H., Chen, J., and Patricio, P., Further results on the inverse along an element in semigroups and rings. Linear Multilinear Algebra 64(2016), 393–403. http://dx.doi.org/10.1080/03081087.2015.1043716 Google Scholar
[12] Zhu, H., Chen, J., and Patricio, P., Reverse order law for the inverse along an element. Linear Multilinear Algebra, to appear. http://dx.doi.org/!0.1080/03081087.2016.1178209 Google Scholar