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

摘要

Let us define A = Circ r ( a 0 , a 1 , … , a n ? 1 ) $A=operatorname{Circ}_{r}(a_{0},a_{1},ldots,a_{n-1})$ to be a n × n $nimes n$ r-circulant matrix. The entries in the first row of A = Circ r ( a 0 , a 1 , … , a n ? 1 ) $A=operatorname{Circ}_{r}(a_{0},a_{1},ldots,a_{n-1})$ are a i = F i $a_{i}=F_{i}$ , or a i = L i $a_{i}=L_{i}$ , or a i = F i L i $a_{i}=F_{i}L_{i}$ , or a i = F i 2 $a_{i}=F_{i}^{2}$ , or a i = L i 2 $a_{i}=L_{i}^{2}$ ( i = 0 , 1 , … , n ? 1 $i=0,1,ldots,n-1$ ), where F i $F_{i}$ and L i $L_{i}$ 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.