On the Bounds for the Norms of Toeplitz Matrices with the Jacobsthal and Jacobsthal Lucas Numbers

Year 2019, Volume: 4 Issue: 3, 105 - 114, 31.12.2019

Şükran Uygun

https://doi.org/10.30931/jetas.569742

https://izlik.org/JA65NP28WR

Abstract

In this study, we compute the value of the various norms of the Toeplitz matrices whose elements are Jacobsthal numbers, Jacobsthal Lucas numbers and upper and lower bounds for the spectral norms of these matrices. Also, the Euclidean norm of Kronecker product of Toeplitz matrices with Jacobsthal and the Jacobsthal Lucas numbers are denoted. Finally, the upper bound for the spectral norm of Hadamard product mentioned above matrices are demonstrated.

Keywords

Jacobsthal , Jacobsthal Lucas sequences , Toeplitz matrices , Hadamard product , Kronecker product

References

• [1] Horadam, A.F., “Jacobsthal representation numbers”, The Fibonacci Quarterly 34(1) (1996) : 40-54.
• [2] Uygun, S., “The (s, t)-Jacobsthal and (s, t)-Jacobsthal Lucas sequences”, Applied Mathematical Sciences 70(9) (2015) : 3467-3476.
• [3] Zielke, G., “Some remarks on matrix norms, condition numbers and error estimates for linear equations”, Linear algebra and its applications 110 (1988) : 29-41.
• [4] Mathias, R., “The spectral norm of a nonnegative matrix”, Linear algebra and its applications 139 (1990) : 269-284.
• [5] Reams, R., “Hadamard inverses, square roots and products of almost semidefinite matrices”, Linear Algebra and its Applications 288 (1999) : 35-43.
• [6] Solak, S., Bozkurt, D., ”On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices”, Applied Mathematics and Computation 140(2-3) (2003) : 231-238.
• [7] Solak, S., “On the norms of circulant matrices with the Fibonacci and Lucas numbers”, Applied Mathematics and Computation 160(1) (2005) : 125-132.
• [8] Horn, R.A., Johnson, C.R., “Topics in matrix analysis”, Cambridge University Press, Cambridge, (1991).
• [9] Daşdemir, A., “On the norms of Toeplitz Matrices with the Pell, Pell-Lucas and Modified Pell numbers”, Journal of Engineering Technology and Applied Sciences 1(2) (2016) : 51-57.
• [10] Shen, S., “On the norms of Toeplitz matrices involving k-Fibonacci and k-Lucas numbers”, Int. J. Contemp. Math. Sciences 7(8) (2012) : 363-368.
• [11] Akbulak, M., Bozkurt, D., “On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers”, Hacettepe Journal of Mathematics and Statistics 37(2) (2008) : 89-95.