On r-circulant matrices with Horadam numbers having arithmetic indices

Aldous Cesar F. Bueno
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 2, Pages 177—197
DOI: 10.7546/nntdm.2020.26.2.177-197
Download full paper: PDF, 238 Kb

Details

Authors and affiliations

Aldous Cesar F. Bueno
Mathematics Unit, Philippine Science High School – Central Luzon Campus
Lily Hill, Clark Freeport Zone, Pampanga, Philippines

Abstract

We investigate an r-circulant matrix whose entries are Horadam numbers having arithmetic indices. We then solve for the eigenvalues, determinant, Euclidean norm and spectral norm of the matrix. Lastly, we present some special cases and some results on identities and divisibility.

Keywords

  • Horadam Numbers
  • r-circulant matrix
  • Eigenvalue
  • Determinant
  • Euclidean norm
  • Spectral norm

2010 Mathematics Subject Classification

  • 11B05
  • 15B36
  • 11B39

References

  1. Bahsi, M., & Solak, S. (2010). On the Circulant Matrices with Arithmetic Sequence, International Journal of Contemporary Mathematical Sciences, 5 (25), 1213–1222.
  2. Bozkurt, D. (2012). On the Determinants and Inverses of Circulant Matrices with a General Number Sequence, Preprint. arXiv: 1202.1068v1 [math.NA] 6 Feb 2012.
  3. Bozkurt, D., & Tam, T. Y. (2012). Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers. Appl. Math. Comput., 219, 544–551.
  4. Bozkurt, D. (2012). On the Determinants and Inverses of Circulant Matrices with Pell and Pell–Lucas Numbers, Preprint. arXiv: 1201.6061v1 [math.NA] 29 Jan 2012.
  5. Bozkurt, D., & Tam, T. Y. (2014). Determinants and inverses of r-circulant matrices associated with a number sequence, Linear and Multilinear Algebra, 63 (10), 2079–2088.
  6. Bueno, A. C. F. (2012). On Right Circulant Matrices with Geometric Progression,
    International Journal of Applied Mathematical Research, 1 (4), 593–603.
  7. Bueno, A. C. F. (2012). Right Circulant Matrices with Fibonacci Sequence, International Journal of Mathematics and Scientific Computing, 2 (2), 9–10.
  8. Bueno, A. C. F. (2013). Right Circulant Matrices with Jacobsthal Sequence, International Journal of Advanced Mathematical Sciences, 1 (2), 87–90.
  9. Bueno, A. C. F. (2013). Generalized Right Circulant Matrices with Geometric Sequence, International Journal of Mathematics and Scientific Computing, 3 (1), 17–18.
  10. Bueno, A. C. F. (2014). On the Eigenvalues and the Determinant of the Right Circulant Matrices with Pell and Pell–Lucas Numbers, International Journal of Mathematics and Scientific Computing, 4 (1), 19–20.
  11. Bueno, A. C. F. (2014). Right Circulant Matrices with Sum of the Terms of Two Geometric Sequences, International Journal of Mathematics and Scientific Computing, 4 (2), 51–52.
  12. Bueno, A. C. F. (2016). Right circulant matrices with ratio of the elements of Fibonacci and geometric sequence, Notes on Number Theory and Discrete Mathematics, 22 (3), 79–83.
  13. Bueno, A. C. F. (2016). Right circulant determinant sequences with Jacobsthal and Jacobsthal–Lucas Numbers, Notes on Number Theory and Discrete Mathematics, 22 (4), 56–61.
  14. Bueno, A. C. F. (2017). On r-circulant matrices with Fibonacci and Lucas numbers having arithmetic indices, AIP Conference Proceedings 1905, Article No. 030010.
  15. Cline, R. E., Plemmons, R. J., & Worm, G. (1974). Generalized Inverses of Certain Toeplitz Matrices, Linear Algebra and Its Applications, 8, 25–33.
  16. Civciv, H., & Turkmen, R. (2008). Notes on Norms of Circulant Matrices with Lucas Numbers, International Journal of Information and System Sciences, 4 (1), 142–147.
  17. Gray, R. (2006). Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Communications and Information Theory, 2 (3), 155–239. Available online at: https://ee.stanford.edu/~gray/toeplitz.pdf.
  18. Horadam, A. F. (1965). Basic Properties of a Certain Generalised Sequence of Numbers, Fibonacci Quarterly 3, 161–176.
  19. Lind, D. A. (1970). A Fibonacci Circulant, Fibonacci Quarterly, 8, 449–455.
  20. Majumdar, A. A. K. (2010). Wandering in the World of Smarandache Numbers, InProQuest, Ann Arbor. Available online at: http://fs.gallup.unm.edu/Majumdar.pdf.
  21. Nalli, A., & Sen, M. (2010). On the Norms of Circulant Matrices with Generalized
    Fibonacci Numbers, Selcuk Journal of Applied Mathematics, 11 (1), 107–116.
  22. Radicic, B. (2019). On k-circulant matrices involving geometric sequence, Hacet. J. Math. Stat., 48 (3), 805–817.
  23. Shen, S., & Cen, J. (2011). On the Norms of Circulant Matrices with the (k,h)-Fibonacci and (k,h)-Lucas Numbers, International Journal of Contemporary Mathematics and Sciences, 6 (19), 887–894.
  24. Shen, S., & Cen, J. (2010). On the Spectral Norms of r-Circulant Matrices with the k-Fibonacci and k-Lucas Numbers, International Journal of Contemporary Mathematics and Sciences, 5 (12), 569–578.
  25. Yalciner, A. (2008). Spectral Norms of Some Special Circulant Matrices, International Journal of Contemporary Mathematics and Sciences, 3 (35), 1733–1738.

Related papers

Cite this paper

Bueno, A. C. F. (2020). On r-circulant matrices with Horadam numbers having arithmetic indices. Notes on Number Theory and Discrete Mathematics, 26 (2), 177-197, doi: 10.7546/nntdm.2020.26.2.177-197.

Comments are closed.