Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SIMPLE FORMULATIONS ON CIRCULANT MATRICES WITH ALTERNATING FIBONACCI

  • Sugi Guritman (Division of Pure Mathematics Department of Mathematics IPB University)
  • Received : 2022.04.23
  • Accepted : 2022.08.17
  • Published : 2023.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this article, an alternating Fibonacci sequence is defined from a second-order linear homogeneous recurrence relation with constant coefficients. Then, the determinant, inverse, and eigenvalues of the circulant matrices with entries in the first row having the formation of the sequence are formulated explicitly in a simple way. In this study, the method for deriving the formulation of the determinant and inverse is simply using traditional elementary row or column operations. For the eigenvalues, the known formulation from the case of general circulant matrices is simplified by considering the specialty of the sequence and using cyclic group properties. We also propose algorithms for the formulation to show how efficient the computations are.

Keywords

References

  1. R. Aldrovandi, Special Matrices of Mathematical Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. https://doi.org/10.1142/9789812799838
  2. M. Bah,si and S. Solak, On the g-circulant matrices, Commun. Korean Math. Soc. 33 (2018), no. 3, 695-704. https://doi.org/10.4134/CKMS.c170244
  3. D. Bozkurt and T.-Y. Tam, Determinants and inverses of r-circulant matrices associated with a number sequence, Linear Multilinear Algebra 63 (2015), no. 10, 2079-2088. https://doi.org/10.1080/03081087.2014.941291
  4. A. C. F. Bueno, Right Circulant Matrices With Geometric Progression, Int. J. Appl. Math. Research 1 (2012), no. 4, 593-603.
  5. A. C. F. Bueno, On r-circulant matrices with Horadam numbers having arithmetic indices, Notes on Number Theory and Discrete Mathematics 26 (2020), no. 2, 177-197. https://doi.org/10.7546/nntdm.2020.26.2.177-197
  6. P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, 1979.
  7. R. P. Grimaldi, Discrete and Combinatorial Mathematics, 4th Edition, Addison Wesley Longman Inc., 1999.
  8. J. Jia and S. Li, On the inverse and determinant of general bordered tridiagonal matrices, Comput. Math. Appl. 69 (2015), no. 6, 503-509. https://doi.org/10.1016/j.camwa.2015.01.012
  9. Z. Jiang, Y. Gong, and Y. Gao, Invertibility and explicit inverses of circulant-type matrices with k-Fibonacci and k-Lucas numbers, Abstr. Appl. Anal. 2014 (2014), Art. ID 238953, 9 pp. https://doi.org/10.1155/2014/238953
  10. X. Jiang and K. Hong, Explicit inverse matrices of Tribonacci skew circulant type matrices, Appl. Math. Comput. 268 (2015), 93-102. https://doi.org/10.1016/j.amc.2015.05.103
  11. Z. Jiang and D. Li, The invertibility, explicit determinants, and inverses of circulant and left circulant and g-circulant matrices involving any continuous Fibonacci and Lucas numbers, Abstr. Appl. Anal. 2014 (2014), Art. ID 931451, 14 pp. https://doi.org/10.1155/2014/931451
  12. P. Lancaster and M. Tismenetsky, The Theory of Matrices, second edition, Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985.
  13. J. Li, Z. Jiang, and F. Lu, Determinants, norms, and the spread of circulant matrices with Tribonacci and generalized Lucas numbers, Abstr. Appl. Anal. 2014 (2014), Art. ID 381829, 9 pp. https://doi.org/10.1155/2014/381829
  14. Z. Liu, S. Chen, W. Xu, and Y. Zhang, The eigen-structures of real (skew) circulant matrices with some applications, Comput. Appl. Math. 38 (2019), no. 4, Paper No. 178, 13 pp. https://doi.org/10.1007/s40314-019-0971-9
  15. J. Ma, T. Qiu, and C. He, A new method of matrix decomposition to get the determinants and inverses of r-circulant matrices with Fibonacci and Lucas numbers, J. Math. 2021 (2021), Art. ID 4782594, 9 pp. https://doi.org/10.1155/2021/4782594
  16. B. Radicic, On k-circulant matrices (with geometric sequence), Quaest. Math. 39 (2016), no. 1, 135-144. https://doi.org/10.2989/16073606.2015.1024185
  17. B. Radicic, On k-circulant matrices involving the Jacobsthal numbers, Rev. Un. Mat. Argentina 60 (2019), no. 2, 431-442. https://doi.org/10.33044/revuma.v60n2a10
  18. S.-Q. Shen, J.-M. Cen, and Y. Hao, On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput. 217 (2011), no. 23, 9790-9797. https://doi.org/10.1016/j.amc.2011.04.072
  19. Y. Wei, Y. Zheng, Z. Jiang, and S. Shon, Determinants, inverses, norms, and spreads of skew circulant matrices involving the product of Fibonacci and Lucas numbers, J. Math. Comput. Sci. 20 (2020), 64-78. https://doi.org/10.22436/jmcs.020.01.08