Skip to main content
SpringerLink
Log in

Solving fuzzy linear systems by a block representation of generalized inverse: the core inverse

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper presents a method for solving fuzzy linear systems, where the coefficient matrix is an \(n\times n\) real matrix, using a block structure of the Core inverse, and we use the Hartwig–Spindelböck decomposition to obtain the Core inverse of the coefficient matrix A. The aim of this paper is twofold. First, we obtain a strong fuzzy solution of fuzzy linear systems, and a necessary and sufficient condition for the existence strong fuzzy solution of fuzzy linear systems are derived using the Core inverse of the coefficient matrix A. Second, general strong fuzzy solutions of fuzzy linear systems are derived, and an algorithm for obtaining general strong fuzzy solutions of fuzzy linear systems by Core inverse is also established. Finally, some examples are given to illustrate the validity of the proposed method.

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.

Similar content being viewed by others

References

  • Abbasbandy S, Otadi M, Mosleh M (2008) Minimal solution of general dual fuzzy linear systems. Chaos Solitons Fractals 37(4):1113–1124

    Article  MathSciNet  Google Scholar 

  • Allahviranloo T (2003) A comment on fuzzy linear systems. Fuzzy Sets Syst 140(3):559–559

    Article  Google Scholar 

  • Allahviranloo T (2004) Numerical methods for fuzzy system of linear equations. Appl Math Comput 155(2):493–502

    MathSciNet  MATH  Google Scholar 

  • Allahviranloo T, Ghanbari M (2012) On the algebraic solution of fuzzy linear systems based on interval theory. Appl Math Model 36(11):5360–5379

    Article  MathSciNet  Google Scholar 

  • Baksalary OM, Trenkler G (2010) Core inverse of matrices. Linear Multilinear Algebra 58(6):681–697

    Article  MathSciNet  Google Scholar 

  • Ben-Israel A, Greville TNE (2003) Generalized inverses: theory and applications. Springer Science and Business Media, Berlin

    MATH  Google Scholar 

  • Berman A, Plemmons R J (1994) Nonnegative matrices in the mathematical sciences. Society for Industrial and Applied Mathematics

  • Friedman M, Ming M, Kandel A (1998) Fuzzy linear systems. Fuzzy Sets Syst 96(2):201–209

    Article  MathSciNet  Google Scholar 

  • Friedman M, Ma M, Kandel A (2003) Author’s reply: “A comment on: ‘fuzzy linear systems’ ”[Fuzzy Sets and Systems 140 (2003), no. 3, 559; MR2035088] by T. Allahviranloo. Fuzzy Sets Syst 140(3): 561–561

  • Hartwig RE, Spindelböck K (1983) Matrices for which \(A^\ast \) and \(A^\dagger \) commute. Linear Multilinear Algebra 14(3):241–256

    Article  MathSciNet  Google Scholar 

  • Lodwick WA, Dubois D (2015) Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets Syst 281:227–251

    Article  MathSciNet  Google Scholar 

  • Ma H, Li T (2019) Characterizations and representations of the core inverse and its applications. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2019.1588847

    Article  Google Scholar 

  • Mihailović B, Jerković VM, Maleśević B (2018) Solving fuzzy linear systems using a block representation of generalized inverses: the Moore–Penrose inverse. Fuzzy Sets Syst 353:44–65

    Article  MathSciNet  Google Scholar 

  • Mihailović B, Jerković VM, Maleśević B (2018) Solving fuzzy linear systems using a block representation of generalized inverses: the group inverse. Fuzzy Sets Syst 353:66–85

    Article  MathSciNet  Google Scholar 

  • Wang H (2011) Rank characterizations of some matrix partial orderings. J East China Normal Univ (Natural Science) 2011(5):5–11

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Wang H, Liu X (2016) Partial orders based on core-nilpotent decomposition. Linear Algebra Appl 488:235–248

    Article  MathSciNet  Google Scholar 

  • Wang H, Zhang X (2019) The core inverse and constrained matrix approximation problem. arXiv:1908.11077

  • Zheng B, Wang K (2006) General fuzzy linear systems. Appl Math Comput 181(2):1276–1286

    MathSciNet  MATH  Google Scholar 

  • Zhou J, Bu C, Wei Y (2012) Group inverse for block matrices and some related sign analysis. Linear Multilinear Algebra 60(6):669–681

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author was supported partially by Innovation Project of Guangxi Graduate Education [No. YCSW2019135], the New Centaury National Hundred, Thousand and Ten Thousand Talent Project of Guangxi [No. GUIZHENGFA210647HAO], the School-level Research Projectin Guangxi University for Nationalities [No. 2018MDQN005], and the Special Fund for Bagui Scholars of Guangxi [No. 2016A17]. The second author was supported partially by Guangxi Natural Science Foundation [No. 2018GXNSFAA138181], by the Xiangsihu Young Scholars Innovative Research Team of Guangxi University for Nationalities [No.GUIKE AD19245148], and the Special Fund for Science and Technological Bases and Talents of Guangxi [No. 2019AC20060]. The third author was supported partially by the National Natural Science Foundation of China [No. 61772006], Guangxi Natural Science Foundation [No. 2018GXNSFDA281023] and the Science and Technology Major Project of Guangxi [No. AA17204096].

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Guangxi University for Nationalities, Nanning, 530006, China

    Hongjie Jiang, Hongxing Wang & Xiaoji Liu

Authors
  1. Hongjie Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongxing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaoji Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongxing Wang.

Ethics declarations

Conflict of interest

No potential conflict of interest was reported by the authors.

Additional information

Communicated by Anibal Tavares de Azevedo.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, H., Wang, H. & Liu, X. Solving fuzzy linear systems by a block representation of generalized inverse: the core inverse. Comp. Appl. Math. 39, 133 (2020). https://doi.org/10.1007/s40314-020-01156-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01156-0

Keywords

Mathematics Subject Classification

Search

Navigation