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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.

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