Inner-gMP and gMP-inner inverses

Year 2024, Volume: 53 Issue: 5, 1312 - 1325, 15.10.2024

Dunja Stojanovic , Dijana Mosic

https://doi.org/10.15672/hujms.1351762

Cited By: 1

Abstract

Solving some systems of operator equations, new kinds of generalized inverses are introduced. Since these new inverses can be expressed by inner and gMP inverses, they are called inner-gMP and gMP-inner inverses. In this way, the concepts of gMP, 1MP and MP1 inverses are generalized. Various representations and characterizations of inner-gMP and gMP-inner inverses are presented. Using the inner and *gMP inverse, we define the inner-*gMP and *gMP-inner inverses which are new extensions of 1MP, MP1 and *gMP inverses. We apply inner-gMP and gMP-inner inverses as well as inner-*gMP and *gMP-inner inverses to solve several kinds of linear equations. Consequently, we obtain solvability of the normal equation which is connected to the least-squares solution. Numerical examples are given to illustrate our results.

Keywords

inner-MP inverse, gMP inverse, core–EP inverse, generalized Drazin inverse

Supporting Institution

Ministry of Education, Science and Technological Development

Project Number

451-03-47/2023-01/200124, 337-00-93/2023-05/13

References

• [1] O.M. Baksalary and G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 (6), 681-697, 2010.
• [2] R. Behera, G. Maharana and J.K. Sahoo, Further results on weighted core-EP inverse of matrices, Results Math. 75, 174, 2020.
• [3] A. Ben-Israel and T.N.E. Greville, Generalized inverses: theory and applications, Second Ed., Springer-Verlag, New York, 2003.
• [4] Y. Chen, K. Zuo and Z. Fu, New characterizations of the generalized Moore-Penrose inverse of matrices, AIMS Mathematics 7 (3), 4359-4375, 2022.
• [5] G. Dolinar, B. Kuzma, J. Marovt and B. Ungor, Properties of core-EP order in rings with involution, Front. Math. China 14, 715-736, 2019.
• [6] D.E. Ferreyra, F.E. Levis and N. Thome, Revisiting the core EP inverse and its extension to rectangular matrices, Quaest. Math. 41 (2), 265-281, 2018.
• [7] Y. Gao and J. Chen, Pseudo core inverses in rings with involution, Comm. Algebra 46 (1), 38-50, 2018.
• [8] M.V. Hernández, M.B. Lattanzi and N. Thome, From projectors to 1MP and MP1 generalized inverses and their induced partial orders, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM 115, 148, 2021.
• [9] Y. Ke, L. Wang and J. Chen, The core inverse of a product and $2\times 2$ matrices, Bull. Malays. Math. Sci. Soc. 42, 51-66, 2019.
• [10] J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38, 367-381, 1996.
• [11] I.I. Kyrchei, Determinantal representations of the core inverse and its generalizations with applications, Journal of Mathematics 2019, Article ID 1631979, 13 pages, 2019.
• [12] K. Manjunatha Prasad and K.S. Mohana, Core–EP inverse, Linear Multilinear Algebra 62 (6), 792-802, 2014.
• [13] J. Marovt, D. Mosić and I. Cremer, On some generalized inverses and partial orders in $\ast $-rings, J. Algebra Appl., 22(12), 2350256, 2023.
• [14] D. Mosić, Weighted core–EP inverse of an operator between Hilbert spaces, Linear Multilinear Algebra 67 (2), 278-298, 2019.
• [15] D. Mosić and D.S. Djordjević, The gDMP inverse of Hilbert space operators, J. Spectr. Theory 8 (2), 555-573, 2018.
• [16] D. Mosić, P.S. Stanimirović, V.N. Katsikis, Solvability of some constrained matrix approximation problems using core-EP inverses, Comput. Appl. Math. 39, 311, 2020.
• [17] D.S. Rakić and M.Z. Ljubenović, 1MP and MP1 inverses and one-sided star orders in a ring with involution, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM 117, 13, 2023.
• [18] K.S. Stojanović and D. Mosić, Generalization of the Moore-Penrose inverse, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114, 196, 2020.
• [19] D. Zhang, Y. Zhao, D. Mosić and V.N. Katsikis, Exact expressions for the Drazin inverse of anti-triangular matrices, J. Comput. Appl. Math. 428, 115187, 2023.
• [20] D. Zhang, Y. Jin and D. Mosić, The Drazin inverse of anti-triangular block matrices, J. Appl. Math. Comput. 68, 2699-2716, 2022.
• [21] M. Zhou, J. Chen and N. Thome, Characterizations and perturbation analysis of a class of matrices related to core-EP inverses, J. Comput. Appl. Math. 393, 113496, 2021.
• [22] H. Zhu and P. Patrício, Characterizations for pseudo core inverses in a ring with involution, Linear Multilinear Algebra 67 (6), 1109-1120, 2019.
• [23] H. Zou, J. Chen and P. Patrício, Reverse order law for the core inverse in rings, Mediterr. J. Math. 15, 145, 2018.