The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers

Abstract

Let us define $A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$ to be a $n\times n$ r-circulant matrix. The entries in the first row of $A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$ are $a_{i}=F_{i}$ , or $a_{i}=L_{i}$ , or $a_{i}=F_{i}L_{i}$ , or $a_{i}=F_{i}^{2}$ , or $a_{i}=L_{i}^{2}$ ( $i=0,1,\ldots,n-1$ ), where $F_{i}$ and $L_{i}$ are the ith Fibonacci and Lucas numbers, respectively. This paper gives an upper bound estimation of the spectral norm for r-circulant matrices with Fibonacci and Lucas numbers. The result is more accurate than the corresponding results of S Solak and S Shen, and of J Cen, and the numerical examples have provided further proof.