Inner-gMP and gMP-inner inverses

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. [1] O.M. Baksalary and G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 (6), 681-697, 2010.
2. [2] R. Behera, G. Maharana and J.K. Sahoo, Further results on weighted core-EP inverse of matrices, Results Math. 75, 174, 2020.
3. [3] A. Ben-Israel and T.N.E. Greville, Generalized inverses: theory and applications, Second Ed., Springer-Verlag, New York, 2003.
4. [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. [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. [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. [7] Y. Gao and J. Chen, Pseudo core inverses in rings with involution, Comm. Algebra 46 (1), 38-50, 2018.
8. [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. [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. [10] J.J. Koliha, A generalized Drazin inverse, Glasgow Math. J. 38, 367-381, 1996.
11. [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. [12] K. Manjunatha Prasad and K.S. Mohana, Core–EP inverse, Linear Multilinear Algebra 62 (6), 792-802, 2014.
13. [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. [14] D. Mosić, Weighted core–EP inverse of an operator between Hilbert spaces, Linear Multilinear Algebra 67 (2), 278-298, 2019.
15. [15] D. Mosić and D.S. Djordjević, The gDMP inverse of Hilbert space operators, J. Spectr. Theory 8 (2), 555-573, 2018.
16. [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. [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. [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. [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. [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. [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. [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. [23] H. Zou, J. Chen and P. Patrício, Reverse order law for the core inverse in rings, Mediterr. J. Math. 15, 145, 2018.