Skip to main content
SpringerLink
Log in

The full rank expressions for the W-weighted Drazin and core-EP inverse of a matrix and their applications

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This paper presents several new expressions for the W-weighted Drazin and core-EP inverse of a matrix based on Urquhart formula. These expressions naturally extend the classic results for the Drazin and core-EP inverse and some of them improve the existing results in the literature. This paper also presents new finite methods for computing the W-weighted Drazin and core-EP inverse through the sequential procedure of full rank factorizations of Cline.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58, 681–697 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Baksalary, O.M., Trenkler, G.: On a generalized core inverse. Appl. Math. Comput. 236, 450–457 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Ben-Israel, A., Greville, T.N.E.: Generalized Inverse: Theory and Applications. Wiley, New York (2003)

    MATH  Google Scholar 

  4. Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. Dover Publication Inc, New York (1991)

    MATH  Google Scholar 

  5. Chen, X., Ji, J.: A finite method for computing the Drazin and core-EP inverses of matrices based on partial full-rank factorization. Commun. Math. Res. 37(4), 448–461 (2021)

    MathSciNet  MATH  Google Scholar 

  6. Cline, R.E.: Inverses of rank invariant powers of a matrix. SIAM J. Numer. Anal. 5, 182–197 (1968)

    MathSciNet  MATH  Google Scholar 

  7. Cline, R.E., Greville, T.N.E.: A Drazin inverse for rectangular matrices. Linear Algebra Appl. 29, 53–62 (1980)

    MathSciNet  MATH  Google Scholar 

  8. Cong, Z., Ma, H.: Characterizations and perturbations of the core-EP inverse of tensors based on the T-product. Numer. Funct. Anal. Optim. 43(10), 1150–1200 (2022)

    MathSciNet  MATH  Google Scholar 

  9. Du, H., Wang, B., Ma, H.: Perturbation theory for core and core-EP inverses of tensor via Einstein product. Filomat 33(16), 5207–5217 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Gao, Y., Chen, J., Patricio, P.: Representations and properties of the weighted core-EP inverse. Linear Multilinear Algebra 68, 1160–1174 (2020)

    MathSciNet  MATH  Google Scholar 

  13. Ji, J., Wei, Y.: The core-EP, weighted core-EP inverse of matrices, and constrained systems of linear equations. Commun. Math. Res. 37(1), 86–112 (2021)

    MathSciNet  MATH  Google Scholar 

  14. Liu, Y., Ma, H.: Perturbation of the weighted T-core-EP inverse of tensors based on the T-product. Commun. Math. Res. 37(4), 496–536 (2021)

    MathSciNet  MATH  Google Scholar 

  15. Liu, Y., Ma, H.: Dual core generalized inverse of third-order dual tensor based on the T-product. Comput. Appl. Math. 41(8), 28 (2022)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Ma, H.: A characterization and perturbation bounds for the weighted core-EP inverse. Quaest. Math. 43, 869–879 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Ma, H.: Perturbation bounds for the core inverse of matrices. Comput. Appl. Math. 41(3), 14 (2022)

    MathSciNet  MATH  Google Scholar 

  19. Ma, H.: Displacement structure of the core inverse. Linear Multilinear Algebra 70(2), 203–214 (2022)

    MathSciNet  MATH  Google Scholar 

  20. Ma, H.: Characterizations and representations for the CMP inverse and its application. Linear Multilinear Algebra 70, 5157–5172 (2022)

    MathSciNet  MATH  Google Scholar 

  21. Ma, H., Gao, X., Stanimirovic, P.S.: Characterizations, iterative method, sign pattern and perturbation analysis for the DMP inverse with its applications. Appl. Math. Comput. 378, 18 (2020)

    MathSciNet  MATH  Google Scholar 

  22. Ma, H., Li, T.T.: Characterizations and representations of the core inverse and its applications. Linear Multilinear Algebra 69(1), 93–103 (2021)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  24. Ma, H., Stanimirovic, P.S., Mosic, D., Kyrchei, I.I.: Sign pattern, usability, representations and perturbation for the core-EP and weighted core-EP inverse. Appl. Math. Comput. 404, 19 (2021)

    MathSciNet  MATH  Google Scholar 

  25. 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 

  26. Mosic, D., Stanimirovic, P.S., Katsikis, V.N.: Solvability of some constrained matrix approximation problems using core-EP inverses. Comput. Appl. Math. 39(4), 21 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Mosic, D., Stanimirovic, P.S., Ma, H.: Generalization of core-EP inverse for rectangular matrices. J. Math. Anal. Appl. 500(1), 19 (2021)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  31. Sahoo, J.K., Behera, R., Stanimirovic, P.S., Katsikis, V.N., Ma, H.: Core and core-EP inverses of tensors. Comput. Appl. Math. 39(1), 28 (2020)

    MathSciNet  MATH  Google Scholar 

  32. Sheng, X.P., Chen, G.: Full-rank representation of generalized inverse \(A_{T, S}^{(2)}\) and its applications. Comput. Math. Appl. 54, 1422–1430 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Sheng, X.P., Xin, D.: Methods of Gauss-Jordan elimination to compute core inverse \(A^{{\bigcirc \!\!\!\!\!\#}}\) and dual core inverse \(A_{{\bigcirc \!\!\!\!\!\#}}\). Linear Multilinear Algebra 70, 2354–2366 (2022)

    MathSciNet  MATH  Google Scholar 

  34. Stanimirovic, P.S., Mosic, D., Wei, Y.: Generalizations of composite inverses with certain image and/or kernel. Appl. Math. Comput. 428, 19 (2022)

    MathSciNet  MATH  Google Scholar 

  35. Stanimirovic, P.S., Pappas, D., Katsikis, V.N., Stanimirovic, I.: Full-rank representations of outer inverses based on the QR decomposition. Appl. Math. Comput. 218, 10321–10333 (2012)

    MathSciNet  MATH  Google Scholar 

  36. Urquhart, N.S.: Computation of generalized inverse matrices which satisfy specified conditions. SIAM Rev. 10, 216–218 (1968)

    MathSciNet  MATH  Google Scholar 

  37. Wang, G., Wei, Y., Qiao, S.: Generalized Inverses: Theory and Computations, 2nd edn. Springer Singapore and Science Press, Beijing (2018)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. Wang, H.X., Chen, J.L., Yan, G.: Generalized Cayley-Hamilton theorem for core-EP inverse matrix and DMP inverse matrix. J. Southeast Uni. 1, 135–138 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Wang, B., Du, H., Ma, H.: Perturbation bounds for DMP and CMP inverses of tensors via Einstein product. Comput. Appl. Math. 39(1), 17 (2020)

    MathSciNet  MATH  Google Scholar 

  41. Wang, H.X., Zhang, X.: The core inverse and constrained matrix approximation problem. Open Math. 18, 653–661 (2020)

    MathSciNet  MATH  Google Scholar 

  42. Wei, Y.: A characterization for the \(W\)-weighted Drazin inverse and Cramer rule for \(W\)-weighted Drazin inverse solution. Appl. Math. Comput. 125, 303–310 (2002)

    MathSciNet  MATH  Google Scholar 

  43. Wei, Y.: Integral representation of the \(W\)-weighted Drazin inverse. Appl. Math. Comput. 144, 3–10 (2003)

    MathSciNet  MATH  Google Scholar 

  44. Wei, Y., Stanimirovic, P.S., Petkovic, M.: Numerical and Symbolic Computations of Generalized Inverses. World Scientific Publishing Co., Singapore (2018)

    MATH  Google Scholar 

  45. 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  MATH  Google Scholar 

  46. Zhou, M., Chen, J., Stanimirovic, P., Katsikis, V., Ma, H.: Complex varying-parameter Zhang neural networks for computing core and core-EP inverse. Neural Process. Lett. 51, 1299–1329 (2020)

    Google Scholar 

Download references

Funding

No funding is available for this research. There are not any financial or non-financial interests that are directly or indirectly related to the work submitted for publication.

Author information

Authors and Affiliations

  1. Department of Mathematics, Kennesaw State University, 1100 S. Marietta Pkwy, Marietta, GA, 30060, USA

    Jun Ji

Authors
  1. Jun Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Ji.

Ethics declarations

Conflict of interest

The author declares that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ji, J. The full rank expressions for the W-weighted Drazin and core-EP inverse of a matrix and their applications. J. Appl. Math. Comput. 69, 2775–2794 (2023). https://doi.org/10.1007/s12190-023-01856-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-023-01856-w

Keywords

Mathematics Subject Classification

Search

Navigation