Skip to main content
SpringerLink
Log in

The Doubly Diagonally Dominant Degree of the Schur Complement of Strictly Doubly Diagonally Dominant Matrices and Its Applications

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

New bounds for the doubly diagonally dominant degree of the Schur complement of strictly doubly diagonally dominant (SDD) matrices are derived and proved to be better than those in Liu et al. (Linear Algebra Appl 437:168–183, 2012). As applications, a new distribution of the eigenvalues and two new infinity norm bounds for the Schur complements of SDD matrices are obtained. Finally, numerical examples are given to verify the theoretical results.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Liu, J.Z., Zhang, J., Liu, Y.: The Schur complement of strictly doubly diagonally dominant matrices and its application. Linear Algebra Appl. 437, 168–183 (2012)

    Article  MathSciNet  Google Scholar 

  2. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)

    Book  Google Scholar 

  3. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979)

    MATH  Google Scholar 

  4. Carlson, D., Markham, T.L.: Schur complements on diagonally dominant matrices. Czechoslov. Math. J. 29(104)(2), 246–251 (1979)

  5. Li, B., Tsatsomeros, M.: Doubly diagonally dominant matrices. Linear Algebra Appl. 261, 221–235 (1997)

    Article  MathSciNet  Google Scholar 

  6. Ikramov, K.D.: Invariance of the Brauer diagonal dominance in Gaussian elimination. Moscow Univ. Comput. Math. Cybern. N2, 91–94 (1989)

    MATH  Google Scholar 

  7. Liu, J.Z., Huang, Y.Q.: Some properties on Schur complements of \(H\)-matrices and diagonally dominant matrices. Linear Algebra Appl. 389, 365–380 (2004)

    Article  MathSciNet  Google Scholar 

  8. Johnson, C.R.: Inverse \(M\)-matrices. Linear Algebra Appl. 47, 195–216 (1982)

    Article  MathSciNet  Google Scholar 

  9. Liu, J.Z., Huang, Y.Q., Zhang, F.Z.: The Schur complements of generalized doubly diagonally dominant matrices. Linear Algebra Appl. 378, 231–244 (2004)

    Article  MathSciNet  Google Scholar 

  10. Liu, J.Z., Zhang, F.Z.: Disc separation of the Schur complements of diagonally dominant matrices and determinantal bounds. SIAM J. Matrix Anal. Appl. 27(3), 665–674 (2005)

    Article  MathSciNet  Google Scholar 

  11. Zhang, C.Y., Li, Y.T., Chen, F.: On Schur complement of block diagonally dominant matrices. Linear Algebra Appl. 414, 533–546 (2006)

    Article  MathSciNet  Google Scholar 

  12. Liu, J.Z., Huang, Z.J.: The Schur complements of \(\gamma \)-diagonally and product \(\gamma \)-diagonally dominant matrix and their disc separation. Linear Algebra Appl. 432, 1090–1104 (2010)

    Article  MathSciNet  Google Scholar 

  13. Li, Y.T., Ouyang, S.P., Cao, S.J., et al.: On diagonal-Schur complements of block diagonally dominant matrices. Appl. Math. Comput. 216(5), 1383–1392 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Zhang, C.Y., Zhu, Y., Luo, S.H., et al.: On generalized Schur complement of nonstrictly diagonally dominant matrices and general \(H\)-matrices. Electron. J. Linear Algebra 23, 801–814 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, C.Y., Xu, F.M., Xu, Z.B., et al.: General \(H\)-matrices and their Schur complements. Front. Math. China 9(5), 1141–1168 (2014)

    Article  MathSciNet  Google Scholar 

  16. Dai, P.F.: A note on diagonal dominance, Schur complements and some classes of \(H\)-matrices and \(P\)-matrices. Adv. Comput. Math. 42(1), 1–4 (2016)

    Article  MathSciNet  Google Scholar 

  17. Zhang, C.Y., Wang, W.W., Luo, S.H., et al.: The disc separation and the eigenvalue distribution of the Schur complement of nonstrictly diagonally dominant matrices. J. Inequal. Appl. 2017, 68 (2017)

    Article  MathSciNet  Google Scholar 

  18. Liu, J.Z., Zhang, J., Zhou, L.X., et al.: The Nekrasov diagonally dominant degree on the Schur complement of Nekrasov matrices and its applications. Appl. Math. Comput. 320, 251–263 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Nedovic, M., Cvetkovic, L.: The Schur complement of \(PH\)-matrices. Appl. Math. Comput. 362, 124541 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Varga, R.S.: Matrix Iterative Analysis, 2nd edn. Springer, Berlin (2000)

    Book  Google Scholar 

  21. Li, G.Q., Liu, J.Z., Zhang, J.: The disc theorem for the Schur complement of two class submatrices with \(\gamma \)-diagonally dominant properties. Numer. Math. Theor. Methods Appl. 10(1), 84–97 (2017)

    Article  MathSciNet  Google Scholar 

  22. Li, C.Q.: Schur complement-based infinity norm bounds for the inverse of SDD matrices. Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-020-00895-x (2020)

Download references

Acknowledgements

The authors are grateful to the anonymous referees for their constructive comments and suggestions. This work is supported by Natural Science Foundation of Guizhou Minzu University (Grant no. GZMU[2019]YB09); The Science and Technology Plan Project of Guizhou Province (Grant no. QKHJC [2017]1084); Science and Technology Top-notch Talents Support Project of Education Department of Guizhou Province (Grant no. QJHKYZ [2016]066); National Natural Science Foundations of China (Grant no. 11501141); West Light Foundation of the Chinese Academy of Sciences.

Author information

Authors and Affiliations

  1. College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, People’s Republic of China

    Jiangliu Gu, Shouwei Zhou, Jianxing Zhao & Jiafeng Zhang

Authors
  1. Jiangliu Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shouwei Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxing Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiafeng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianxing Zhao.

Additional information

Communicated by Ali Armandnejad.

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

Gu, J., Zhou, S., Zhao, J. et al. The Doubly Diagonally Dominant Degree of the Schur Complement of Strictly Doubly Diagonally Dominant Matrices and Its Applications. Bull. Iran. Math. Soc. 47, 265–285 (2021). https://doi.org/10.1007/s41980-020-00382-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-020-00382-w

Keywords

Mathematics Subject Classification

Search

Navigation