On Geometric Circulant Matrices Whose Entries are Bi-Periodic Fibonacci and Bi-Periodic Lucas Numbers

Year 2020, Volume: 3 Issue: 3, 102 - 108, 29.09.2020

Emrah Polatlı

https://doi.org/10.32323/ujma.669276

Cited By: 3

Abstract

In this study, we obtain upper and lower bounds for the spectral norms of the geometric circulant matrices with the bi--periodic Fibonacci numbers and bi--periodic Lucas numbers, respectively. Then we give some bounds for the spectral norms of Kronecker and Hadamard products of these matrices.                                                                                                                                                                                                                  

Keywords

Bi-periodic Fibonacci numbers, Bi-periodic Lucas numbers, Geometric circulant matrices, Spectral norm

References

• [1] M. Bahşi, S. Solak, On the norms of $\mathit{r}$–circulant matrices with the hyper-Fibonacci and Lucas numbers, J. Math. Inequal., 8(4) (2014) 693–705.
• [2] M. Bahşi, On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers, TWMS J. Pure Appl. Math., 6(1) (2015), 84–92.
• [3] M. Bahşi, On the norms of $\mathit{r}$–circulant matrices with the hyperharmonic numbers, J. Math. Inequal., 10(2) (2016), 445–458.
• [4] G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput., 245 (2014), 526–538.
• [5] M. Edson, O. Yayenie, A new generalization of Fibonacci sequence and extended Binet’s formula, INTEGERS, 9 (2009), 639–654.
• [6] C. He, J. Ma, K. Zhang, Z. Wang, The upper bound estimation on the spectral norm of $\mathit{r}$-circulant matrices with the Fibonacci and Lucas numbers, J. Inequal. Appl., 2015 (2015), Article ID 72, 10 pages.
• [7] R. Horn, C. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
• [8] Z. Jiang, Z. Zhou, A note on spectral norms of even-order $\mathit{r}$-circulant matrices, Appl. Math. Comput., 250 (2015), 368–371.
• [9] C. Kızılateş, On the Quadra Lucas-Jacobsthal numbers, Karaelmas Fen ve Müh. Derg., 7(2) (2017), 619–621.
• [10] C. Kızılateş, N. Tuğlu, On the bounds for the spectral norms of geometric circulant matrices, J. Inequal. Appl., 2016 (2016), Article ID 312, 15 pages.
• [11] C. Kızılates¸, N. Tu˘glu, On the Norms of Geometric and Symmetric Geometric Circulant Matrices with the Tribonacci Number, Gazi Univ. J. Sci., 31(2) (2018), 555–567.
• [12] E. G. Koçer, T. Mansour, N. Tuğlu, Norms of circulant and semicirculant matrices with Horadams’s numbers, Ars Comb., 85 (2007), 353–359.
• [13] C. Köme, Y. Yazlık, On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers, J. Inequal. Appl., 2017 (2017), Article ID 192, 12 pages.
• [14] R. Mathias, The spectral norm of a nonnegative matrix, Linear Algebra Appl., 139 (1990), 269-284.
• [15] B. Radicic, On k–circulant matrices (with geometric sequence), Quaest. Math., 39 (2016), 135-144.
• [16] S. Q. Shen, J. M. Cen, On the bounds for the norms of $\mathit{r}$–circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput., 216 (2010), 2891–2897.
• [17] S. Q. Shen, J. M. Cen, On the Spectral Norms of $\mathit{r}$-Circulant Matrices with the k -Fibonacci and k-Lucas Numbers, Int. J. Contemp. Math. Sci., 5(12) (2010), 569–578.
• [18] B. Shi, The spectral norms of geometric circulant matrices with the generalized k-Horadam numbers, J. Inequal. Appl., 2018 (2018), Article ID 14, 9 pages.
• [19] W. Sintunavarat, The upper bound estimation for the spectral norm of $\mathit{r}$–circulant and symmetric $\mathit{r}$–circulant matrices with the Padovan sequence, J. Nonlinear Sci. Appl., 9 (2016), 92–101.
• [20] S. Solak, On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 160 (2005), 125–132.
• [21] N. Tuğlu, C. Kızılates¸, On the norms of circulant and $\mathit{r}$–circulant matrices with the hyperharmonic Fibonacci numbers, J. Inequal. Appl., 2015 (2015), Article ID 253, 11 pages.
• [22] N. Tuğlu, C. Kızılateş, On the norms of some special matrices with the harmonic Fibonacci numbers, Gazi Univ. J. Sci., 28(3) (2015), 497–501.
• [23] N. Tuğlu, C. Kızılateş, S. Kesim, On the harmonic and hyperharmonic Fibonacci numbers, Adv. Difference Equ., 2015 (2015), Article ID 297, 12 pages.
• [24] O. Yayenie, New identities for generalized Fibonacci sequences and new generalization of Lucas sequences, SEA Bull. Math., 36 (2012), 739–752.
• [25] Y. Yazlık, N. Taşkara, On the norms of an r–circulant matrix with the generalized k–Horadam numbers, J. Inequal. Appl., 2013 (2013), Article ID 394, 8 pages.