On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers

Abstract

In this paper, we compute the spectral norms of $r-$ circulant matrices with the hyper-Fibonacci and hyper-Lucas numbers of the forms $F_{r}=Circ-r(F_k^{(0)},F_k^{(1)},...,F_k^{(n-1)}) $, $L_r=Circ-r(L_{k}^{\left( 0\right) },L_{k}^{\left( 1\right) },...,L_{k}^{\left( n-1\right) })$ and their Hadamard and Kronecker products. For this, we firstly compute the spectral and Euclidean norms of circulant matrices of the forms $F=Circ(F_{k}^{\left( 0\right) }, F_{k}^{\left( 1\right) },... ,F_{k}^{\left( n-1\right) })$ and $L=Circ(L_{k}^{\left( 0\right) },L_{k}^{\left( 1\right) },...,L_{k}^{\left( n-1\right) })$. Moreover, we give some examples related to special cases of our results.