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Representations and properties for the MPCEP inverse

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Abstract

Our contribution is the development of novel representations and investigations of main properties of the MPCEP inverse. Precisely, we present representations of the MPCEP inverse which involve appropriate Moore–Penrose inverses, projections and full-rank decompositions, as well as limit and integral representations. Determinantal representations for the MPCEP inverse are also established. We study perturbation formulae with perturbation bounds of MPCEP inverse. An equivalent condition for the continuity of the MPCEP inverse is proposed. The MPCEP inverse of a suitable upper block triangular matrix is given. The successive matrix squaring algorithm and splitting method for computing the MPCEP inverse are presented. Some appropriate constrained systems of linear equations are solved applying the MPCEP inverse.

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References

  1. Ben-Israel, A., Greville, T.N.E.: Generalized inverses. Theory and Applications, 2nd edn. In: Canadian Mathematical Society. Springer, New York (2003)

  2. Benítez, J., Boasso, E., Jin, H.: On one-sided \((B, C)\)-inverses of arbitrary matrices. Electron. J. Linear Algebra 32, 391–422 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Berman, A., Neumann, M.: Proper splittings of rectangular matrices. SIAM J. Appl. Math. 31, 307–312 (1976)

    MathSciNet  MATH  Google Scholar 

  4. Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. Pitman, London (1979)

    MATH  Google Scholar 

  5. Chen, J.L., Mosić, D., Xu, S.Z.: On a new generalized inverse for Hilbert space operators. Quaest. Math. 43(9), 1331–1348 (2020)

    MathSciNet  MATH  Google Scholar 

  6. Chen, J.L., Xu, S.Z., Benítez, J., Chen, X.F.: Rank equalities related to a class of outer generalized inverse. Filomat 33(17), 5611–5622 (2019)

    MathSciNet  Google Scholar 

  7. Castro-González, N., Koliha, J.J., Wei, Y.: On integral representations of the Drazin inverse in Banach algebras. Proc. Edinb. Math. Soc. 45, 327–331 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Castro-González, N., Koliha, J.J., Wei, Y.: Integral representation of the Drazin inverse \(A^D\). Electron. J. Linear Algebra 9, 129–131 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Deng, C.Y., Du, H.K.: Representations of the Moore–Penrose inverse of \(2\times 2\) block operator valued matrices. J. Korean Math. Soc. 46(6), 1139–1150 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Ferreyra, D.E., Levis, F.E., Thome, N.: Maximal classes of matrices determining generalized inverses. Appl. Math. Comput. 333, 42–52 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Ferreyra, D.E., Levis, F.E., Thome, N.: Revisiting the core EP inverse and its extension to rectangular matrices. Quaest. Math. 41(2), 265–281 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Gao, Y.: Perturbation bounds for the W-weighted core-EP inverse. J. Algebra Appl. 19(12), 2050232 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Gao, Y., Chen, J.: Pseudo core inverses in rings with involution. Commun. Algebra 46(1), 38–50 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Gao, Y., Chen, J., Patricio, P.: Continuity of the core-EP inverse and its applications. Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2019.1608899

  15. Groetsch, C.W.: Generalized inverses of linear operators: representation and approximation. In: Monographs and Textbooks in Pure and Applied Mathematics, Vol. 37. Marcel Dekker Inc., New York (1977)

  16. Kyrchei, I.: Determinantal representations of the core inverse and its generalizations with applications. J. Math. 2019, Article ID 1631979 (2019)

  17. Kyrchei, I.: Analogs of the adjoint matrix for generalized inverses and corresponding Cramer rules. Linear Multilinear Algebra 56(4), 453–469 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Kyrchei, I.: Weighted quaternion core-EP, DMP, MPD, and CMP inverses and their determinantal representations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114, 198 (2020). https://doi.org/10.1007/s13398-020-00930-3

  19. Liu, X., Cai, N.: High-order iterative methods for the DMP inverse. J. Math. 2018, Article ID 8175935

  20. Liu, X., Huang, S.: Proper splitting for the generalized inverse \(A^{(2)}_{T,S}\) and its application on Banach spaces. Abstr. Appl. Anal. 2012, Article ID 736929

  21. Liu, X., Yu, Y., Zhong, J., Wei, Y.: Integral and limit representations of the outer inverse in Banach space. Linear Multilinear Algebra 60, 333–347 (2012)

    MathSciNet  MATH  Google Scholar 

  22. Meng, L.S.: The DMP inverse for rectangular matrices. Filomat 31(19), 6015–6019 (2017)

    MathSciNet  Google Scholar 

  23. Ma, H.: Optimal perturbation bounds for the core inverse. Appl. Math. Comput. 336, 176–181 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Ma, H., Stanimirović, P.S.: Characterizations, approximation and perturbations of the core-EP inverse. Appl. Math. Comput. 359, 404–417 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Ma, H., Gao, X., Stanimirović, P.S.: Characterizations, iterative method, sign pattern and perturbation analysis for the DMP inverse with its applications. Appl. Math. Comput. 378, 125196 (2020)

    MathSciNet  MATH  Google Scholar 

  26. Malik, S.B., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226, 575–580 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Meyer, C.D.: Limits and the index of a square matrix. SIAM J. Appl. Math. 26, 469–478 (1974)

    MathSciNet  MATH  Google Scholar 

  28. Mosić, D.: Core-EP inverses in Banach algebras. Linear Multilinear Algebra (2019). https://doi.org/10.1080/03081087.2019.1701976

  29. Mosić, D.: Core-EP inverse in rings with involution. Publ. Math. Debrecen 96(3–4), 427–443 (2020)

    MathSciNet  MATH  Google Scholar 

  30. Mosić, D.: Perturbation of the weighted core-EP inverse. Ann. Funct. Anal. 11, 75–86 (2020)

    MathSciNet  MATH  Google Scholar 

  31. Mosić, D.: Weighted gDMP inverse of operators between Hilbert spaces. Bull. Korean Math. Soc. 55, 1263–1271 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Mosić, D.: Weighted core-EP inverse of an operator between Hilbert spaces. Linear Multilinear Algebra 67(2), 278–298 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Mosić, D.: Weighted core-EP inverse and weighted core-EP pre-orders in a \(C^*\)-algebra. J. Aust. Math. Soc. (2019). https://doi.org/10.1017/S1446788719000387

  34. Mosić, D., Djordjević, D.S.: The gDMP inverse of Hilbert space operators. J. Spectr. Theor. 8(2), 555–573 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Mosić, D., Stanimirović, P.S.: Composite outer inverses for rectangular matrices. Quaest. Math. https://doi.org/10.2989/16073606.2019.1671526

  36. Nandi, A.K., Sahoo, J.K., Ghosh, P.: Three-step alternating and preconditioned scheme for rectangular matrices. J. Appl. Math. Comput. 60, 485–515 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Prasad, K.M.: An introduction to generalized inverse. In: Bapat, R.B., Kirkland, S., Manjunatha Prasad, K., Puntanen, S. (eds.) Lectures on matrix and graph methods, pp. 43–60. Manipal University Press, Manipal (2012)

    Google Scholar 

  38. Prasad, K.M., Mohana, K.S.: Core-EP inverse. Linear Multilinear Algebra 62(6), 792–802 (2014)

    MathSciNet  MATH  Google Scholar 

  39. Prasad, K.M., Raj, M.D.: Bordering method to compute core-EP inverse. Spec. Matr. 6, 193–200 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Prasad, K.M., Raj, M.D., Vinay, M.: Iterative method to find core-EP inverse. Bull. Kerala Math. Assoc. Spec. Issue 16(1), 139–152 (2018)

    MathSciNet  Google Scholar 

  41. Sahoo, J.K., Behera, R., Stanimirović, P.S., Katsikis, V.N., Ma, H.: Core and core-EP inverses of tensors. Comput. Appl. Math. 39, 9 (2020). https://doi.org/10.1007/s40314-019-0983-5

    Article  MathSciNet  MATH  Google Scholar 

  42. Stanimirović, P.S.: Limit representations of generalized inverses and related methods. Appl. Math. Comput. 103, 51–68 (1999)

    MathSciNet  MATH  Google Scholar 

  43. Srivastava, S., Gupta, D.K.: A higher order iterative method for \(A^{(2)}_{T, S}\). J. Appl. Math. Comput. 46, 147–168 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Stanimirović, P.S., Cvetković-Ilić, D.S.: Successive matrix squaring algorithm for computing outer inverses. Appl. Math. Comput. 203, 19–29 (2008)

    MathSciNet  MATH  Google Scholar 

  45. Stewart, G.W.: On the continuity of the generalized inverse. SIAM J. Appl. Math. 17, 33–45 (1969)

    MathSciNet  MATH  Google Scholar 

  46. Wang, B., Du, H., Ma, H.: Perturbation bounds for DMP and CMP inverses of tensors via Einstein product. Comput. Appl. Math. 39, 28 (2020). https://doi.org/10.1007/s40314-019-1007-1

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang, H.: Core-EP decomposition and its applications. Linear Algebra Appl. 508, 289–300 (2016)

    MathSciNet  MATH  Google Scholar 

  48. Wang, G., Wei, Y., Qiao, S.: Generalized inverses: theory and computations. In: Developments in Mathematics, vol. 53. Springer, Beijing (2018)

  49. Wei, Y.: A characterization and representation of the generalized inverse \(A^{(2)}_{T, S}\) and its applications. Linear Algebra Appl. 280, 79–86 (1998)

    Google Scholar 

  50. Wei, Y.: Index splitting for the Drazin inverse and the singular linear system. Appl. Math. Comput. 95, 115–124 (1998)

    MathSciNet  MATH  Google Scholar 

  51. Yu, A., Deng, C.: Characterization of DMP inverse in Hilbert space. Calcolo 53, 331–341 (2016)

    MathSciNet  MATH  Google Scholar 

  52. Zhou, M., Chen, J.: Integral representations of two generalized core inverses. Appl. Math. Comput. 333, 187–193 (2018)

    MathSciNet  MATH  Google Scholar 

  53. Zhou, M.M., Chen, J.L., Li, T.T., Wang, D.G.: Three limit representations of the core-EP inverse. Filomat 32, 5887–5894 (2018)

    MathSciNet  Google Scholar 

  54. Zhu, H.: On DMP inverses and m-EP elements in rings. Linear Multilinear Algebra 67(4), 756–766 (2019)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Mathematics, University of Niš, Višegradska 33, Niš, 18000, Serbia

    Dijana Mosić

  2. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, 3-b, Naukova Str., L’viv, 79060, Ukraine

    Ivan I. Kyrchei

  3. Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, Niš, 18000, Serbia

    Predrag S. Stanimirović

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  1. Dijana Mosić
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  2. Ivan I. Kyrchei
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  3. Predrag S. Stanimirović
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Dijana Mosić is supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174007 (451-03-68/2020-14/200124). This author also acknowledge the bilateral project between Serbia and Slovenia (Generalized inverses, operator equations and applications) was financially supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 337-00-21/2020-09/32.

Predrag Stanimirović acknowledges support from the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174013 (451-03-68/2020-14/200124).

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Mosić, D., Kyrchei, I.I. & Stanimirović, P.S. Representations and properties for the MPCEP inverse. J. Appl. Math. Comput. 67, 101–130 (2021). https://doi.org/10.1007/s12190-020-01481-x

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