On the bounds for the norms of circulant matrices with the Jacobsthal and Jacobsthal–Lucas numbers

Ş. Uygun and S. Yaşamalı
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 1, Pages 91–98
Full paper (PDF, 175 Kb)

Details

Authors and affiliations

Ş. Uygun
Department of Mathematics, Science and Art Faculty
Gaziantep University, Campus, 27310, Gaziantep, Turkey

S. Yaşamalı
Department of Mathematics, Science and Art Faculty
Gaziantep University, Campus, 27310, Gaziantep, Turkey

Abstract

In this study, we have found upper and lower bounds for the spectral norms of circulant matrices in the forms A = C_r(j₀, j₁, …, j_n−1) and B = C_r(c₀, c₁, …, c_n−1).

Keywords

• Jacobsthal number
• Jacobsthal–Lucas number
• Circulant matrix
• Norm

AMS Classification

• 15A36
• 15A45
• 15A60

References

1. Horadam, A. F. (1996) Jacobsthal Representation Numbers, The Fibonacci Quarterly, 34(1), 40–54.
2. Koshy, T. (2001) Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc., New York.
3. Sloane, N. J. A. (2006) The on-line Encyclopedia of Integer Sequences.
4. Zielke, G. (1988) Some remarks on matrix norms, condition numbers and error estimates for linear equations, Linear Algebra and its Applications, 110, 29–41.
5. Mathias, R. (1990) The spectral norm of nonnegative matrix, Linear Algebra and its Applications, 131, 269–284.
6. Reams, R. (1999) Hadamard inverses square roots and products of almost semidefinite matrices, Linear Algebra and its Applications, 288, 35–43.
7. Solak, S. & Bozkurt, D. (2003) On the spectral norms of Cauchy–Toeplitz and Cauchy– Henkel matrices, Appl. Math. Comput., 140, 231–238.
8. Solak, S. (2005) On the norms of circulant matrices with the Fibonacci and Lucas numbers, Applied Mathematics and Computation, 160, 125–132.
9. Shen, S. & Cen, J. (2010) On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 216, 2891–2897.
10. Shen, S. (2010) On the spectral norms of r-circulant matrices with the k-Fibonacci and k-Lucas numbers, Int. J. Contemp. Math. Sciences, 5(12) 569–578.
11. Horn, R. A. & Johnson, C. R. (1991) Topics in Matrix Analysis, Cambridge University Press, Cambridge.
12. Uygun, Ş. (2015) The (s, t)-Jacobsthal and (s, t)-Jacobsthal–Lucas sequences, Applied Mathematical Sciences, 70(9), 3467–3476.
13. Uslu, K., Taskara, N. & Uygun, Ş. (2011) The relations among k-Fibonacci, k-Lucas and generalized k-Fibonacci numbers and the spectral norms of the matrices of involving these numbers, Ars Combinatoria, 102, 183–192.