On the norms of r-circulant matrices with generalized Fibonacci numbers

Year 2017, Volume: 4 Issue: 1, 13 - 21, 11.01.2017

Amara Chandoul

https://doi.org/10.13069/jacodesmath.12813

Cited By: 3

Abstract

In this paper, we obtain a generalization of [6, 8]. Firstly, we consider the so-called r-circulant matrices
with generalized Fibonacci numbers and then found lower and upper bounds for the Euclidean
and spectral norms of these matrices. Afterwards, we present some bounds for the spectral norms of
Hadamard and Kronecker product of these matrices.

Keywords

Matrix norm, r-circulant matrices, Generalized Fibonacci numbers

References

• [1] D. Bozkurt, T. Y. Tam, Determinants and inverses of r􀀀circulant matrices associated with a number sequences, Linear and Multilinear Algebra 63(10) (2015) 2079–2088.
• [2] H. Civciv, R. Turkmen, Notes on norms of circulant matrices with Lucas numbers, Int. J. Inf. Syst. Sci. 4(1) (2008) 142–147.
• [3] C. He, J. Ma, K. Zhang, Z. Wang, The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers, J. Inequal. Appl. Article ID 72 (2015) 1–10.
• [4] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
• [5] R. Mathias, The spectral norm of nonnegative matrix, Linear Algebra Appl. 139 (1990), 269–284.
• [6] A. Nalli, M. Sen, On the norms of circulant matrices with generalized Fibonacci numbers, Selçuk J. Appl. Math. 11(1) (2010) 107–116.
• [7] T. Koshy, Fibonacci and Lucas Numbers with Application, John Wiley and Sons, Inc., 2001.
• [8] S. Shen, J. Cen, On the norms of r􀀀circulant matrices with Fibonacci and Lucas numbers, Appl.
• [9] S. Solak, On the norms of circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput. 160(1) (2005) 125–132.
• [10] S. Solak, Erratum to "On the norms of circulant matrices with Fibonacci and Lucas numbers" [Appl. Math. Comput. 160(1) (2005) 125-132], Appl. Math. Comput. 190(2) (2007) 1855–1856.
• [11] N. Tuglu, C. Kizilates, On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers, J. Inequal. Appl. Article ID 253 (2015) 1–11.
• [12] S. Vajda, Fibonacci and Lucas numbers and Golden Section, John Wiley and Sons, Inc., 1989.