Vol 19 No 4 (2025): BAREKENG: Journal of Mathematics and Its Application
Articles

ON CONDITIONS FOR MATRICES T SUCH THAT T-I AND I-T^(-1) ARE INVERSE H-MATRICES*

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  • Jeriko Gormantara,
  • Hanni Garminia,
  • Amir Kamal Amir,
  • Evan Ramdan
Jeriko Gormantara
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Hasanuddin, Indonesia
Hanni Garminia
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia
Amir Kamal Amir
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Hasanuddin, Indonesia
Evan Ramdan
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Hasanuddin, Indonesia
DOI: https://doi.org/10.30598/barekengvol19iss4pp2953-2962
Published September 1, 2025
Keywords
  • Group Inverse,
  • H-Matrix,
  • Inverse H-Matrix,
  • M-Matrix
How to Cite
[1]
J. Gormantara, H. Garminia, A. K. Amir, and E. Ramdan, “ON CONDITIONS FOR MATRICES T SUCH THAT T-I AND I-T^(-1) ARE INVERSE H-MATRICES*”, BAREKENG: J. Math. & App., vol. 19, no. 4, pp. 2953-2962, Sep. 2025.

Copyright (c) 2025 Jeriko Gormantara, Hanni Garminia, Amir Kamal Amir, Evan Ramdan

Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Abstract

In this article, we study an analogue of a classical result for M-matrices: if T - I is an invertible M-matrix, then both T and I - T-1 are also invertible M-matrices. We extend this implication to a broader class—inverse H-matrices. The expressions T - I and I - T-1 commonly arise in the analysis of matrix stability, convergence of iterative methods, and spectral transformations, making their structural properties important for numerical analysis. We demonstrate that this implication does not generally hold for inverse H-matrices. However, we derive some conditions under which it remains valid. Specifically, we prove that under certain conditions, if T - I  is an inverse H-matrix, then T and I - T-1 are also inverse H-matrices. Additionally, we investigate the result in the context of group inverses, showing that it does not hold for group inverse M-matrices and H-matrices.

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References

  1. R. Bru, M. T. Gassó, I. Giménez, and J. A. Scott, “THE HADAMARD PRODUCT OF A NONSINGULAR GENERAL H-MATRIX AND ITS INVERSE TRANSPOSE IS DIAGONALLY DOMINANT,” J Appl Math, vol. 2015, 2015, doi: https://doi.org/10.1155/2015/264680.
  2. J. Zhao, Q. Liu, C. Li, and Y. Li, “DASHNIC–ZUSMANOVICH TYPE MATRICES: A NEW SUBCLASS OF NONSINGULAR H-MATRICES,” Linear Algebra Appl, vol. 552, pp. 277–287, Sep. 2018, doi: ttps://doi.org/10.1016/j.laa.2018.04.028.
  3. M. Chatterjee and K. C. Sivakumar, “Inverse H-matrices,” Linear Algebra Appl, vol. 572, pp. 1–31, 2019, doi: https://doi.org/10.1016/j.laa.2019.02.019
  4. J. J. McDonald et al., “M-MATRIX AND INVERSE M-MATRIX EXTENSIONS,” Special Matrices, vol. 8, no. 1, pp. 186–203, Jan. 2020, doi: https://doi.org/10.1515/spma-2020-0113
  5. M. Chatterjee and K. C. Sivakumar, “INTERVALS OF H-MATRICES AND INVERSE M-MATRICES,” Linear Algebra Appl, vol. 614, pp. 24–43, Apr. 2021, doi: https://doi.org/10.1016/j.laa.2019.12.001.
  6. S. Mondal, K. C. Sivakumar, and M. Tsatsomeros, “NEW RESULTS ON M-MATRICES, H-MATRICES AND THEIR INVERSE CLASSES,” Electronic Journal of Linear Algebra, vol. 38, pp. 729–744, 2022, doi: https://doi.org/10.13001/ela.2022.7177.
  7. D. Cvetković, Đ. Vukelić, and K. Doroslovački, “A New Subclass of H-Matrices with Applications,” Mathematics, vol. 12, no. 15, Aug. 2024, doi: https://doi.org/10.3390/math12152322
  8. F. Luo, W. Hu, E. Wu, and X. Yuan, “GLOBAL EXPONENTIAL STABILITY OF IMPULSIVE DELAYED NEURAL NETWORKS WITH PARAMETER UNCERTAINTIES AND REACTION–DIFFUSION TERMS,” Mathematics, vol. 12, no. 15, Aug. 2024, doi: https://doi.org/10.3390/math12152395.
  9. A. M. Encinas, K. Kranthi Priya, and K. C. Sivakumar, “Tridiagonal M-matrices whose group inverses are tridiagonal,” Linear Algebra Appl, vol. 708, pp. 42–60, 2025, doi: https://doi.org/10.1016/j.laa.2024.11.026
  10. S. M. A.M. Encinas and K. C. Sivakumar, “ON AN ANALOGUE OF A PROPERTY OF SINGULAR M-MATRICES FOR THE LYAPUNOV AND STEIN OPERATORS,” Linear and Multilinear Algebra, vol. 73, no. 1, pp. 1–16, 2025, doi: https://doi.org/10.1080/03081087.2024.2313633.
  11. S. Wang, Y. Shi, and J. Guo, “EXPONENTIAL STABILITY OF A CLASS OF QUATERNION-VALUED MEMRISTOR-BASED NEURAL NETWORK WITH TIME-VARYING DELAY VIA M-MATRIX,” Math Methods Appl Sci, vol. 48, no. 3, pp. 3304–3315, 2025, doi: https://doi.org/10.1002/mma.10486
  12. L. Dong, J. Li, and G. Li, “THE DOUBLE DEFLATING TECHNIQUE FOR IRREDUCIBLE SINGULAR M-MATRIX ALGEBRAIC RICCATI EQUATIONS IN THE CRITICAL CASE,” Linear and Multilinear Algebra, vol. 67, no. 8, pp. 1653–1684, 2019, doi: https://doi.org/10.1080/03081087.2018.1466862.
  13. L. Dong and J. Li, “A CLASS OF COMPLEX NONSYMMETRIC ALGEBRAIC RICCATI EQUATIONS ASSOCIATED WITH H-MATRIX,” J Comput Appl Math, vol. 368, p. 112567, 2020, doi: https://doi.org/10.1016/j.cam.2019.112567
  14. C.-H. Guo, “ON ABSOLUTE VALUE EQUATIONS ASSOCIATED WITH M-MATRICES AND H-MATRICES,” Appl Math Lett, vol. 166, p. 109550, 2025, doi: https://doi.org/10.1016/j.aml.2025.109550
  15. J. Guan and Z. Wang, “NUMERICAL SOLUTION OF A CLASS OF QUADRATIC MATRIX EQUATIONS,” Filomat, vol. 39, pp. 33–40, 2025, doi: 10.2298/FIL2501033G.
  16. Q. Zhong, “LOWER BOUND FOR THE MINIMAL EIGENVALUE OF M -MATRICES,” J Phys Conf Ser, vol. 2964, no. 1, p. 012045, Feb. 2025, doi: https://doi.org/10.1088/1742-6596/2964/1/012045
  17. L.-X. Wang, Y. Cao, and Q.-Q. Shen, “IMPROVED MODULUS-BASED MATRIX SPLITTING ITERATION METHODS FOR A CLASS OF HORIZONTAL IMPLICIT COMPLEMENTARITY PROBLEMS,” J Comput Appl Math, vol. 454, p. 116183, 2025, doi: https://doi.org/10.1016/j.cam.2024.116183
  18. M. Nedović and D. Arsić, “NEW SCALING CRITERIA FOR H-MATRICES AND APPLICATIONS,” AIMS Mathematics, vol. 10, no. 3, pp. 5071–5094, 2025, doi: https://doi.org/10.3934/math.2025232
  19. Y. Liu, S. Wu, and C. Li, “IMPROVED CONVERGENCE THEOREM FOR THE GENERAL MODULUS-BASED MATRIX SPLITTING METHOD,” Journal of Applied Analysis and Computation, vol. 15, no. 2, pp. 951–957, 2025, doi: https://doi.org/10.11948/20240215
  20. C. Ma, Y. Wu, and Y. Xie, “The newton-based matrix splitting iterative method for solving generalized absolute value equation with nonlinear term *,” Journal of Applied Analysis and Computation, vol. 15, no. 2, pp. 896–914, 2025, doi: https://doi.org/10.11948/20240198
  21. L. Yu. Kolotilina, “SSDD MATRICES AND RELATIONS WITH OTHER SUBCLASSES OF THE NONSINGULAR ℋ-MATRICES,” Journal of Mathematical Sciences, 2025, doi: https://doi.org/10.1007/s10958-025-07711-6
  22. W. Zeng, Q.-W. Wang, and J. Liu, “CONSTRUCTION AND DECOMPOSITION OF SCALING MATRICES FOR S^k-SDD MATRICES AND THEIR APPLICATION TO LINEAR COMPLEMENTARITY PROBLEMS,” Numer Algorithms, 2025, doi: https://doi.org/10.1007/s11075-025-02047-3
  23. A. Kalauch, S. Lavanya, and K. C. Sivakumar, “SINGULAR IRREDUCIBLE M-MATRICES REVISITED,” Linear Algebra Appl, vol. 565, pp. 47–64, 2019, doi: https://doi.org/10.1016/j.laa.2018.11.030
  24. M. Chatterjee and K. C. Sivakumar, “INEQUALITIES FOR GROUP INVERTIBLE H-MATRICES,” Linear Algebra Appl, vol. 576, no. April, pp. 158–180, 2019, doi: https://doi.org/10.1016/j.laa.2018.04.011
  25. A. Kalauch, S. Lavanya, and K. C. Sivakumar, “Matrices whose group inverses are M-matrices,” Linear Algebra Appl, vol. 614, pp. 44–67, 2021, doi: https://doi.org/10.1016/j.laa.2019.12.026