On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers

In this paper, we present new upper and lower bounds for the spectral norms of the r-circulant matrices Q = C-r ((b/a)(xi(1)/2) q(0), (b/a)(xi(2)/2) q(1), (b/a)(xi(3)/2) q(2) ,..., (b/a)(xi(n)/2) q(n-1)) and L = C-r ((b/a)(xi(0)/2)/(0), (b/a)(xi(1)/2)/1,(b/a)(xi(2)/2)/2 ,..., (b/a)(xi(n-1)/2)/(n-1)) whose entries are the biperiodic Fibonacci and biperiodic Lucas numbers, respectively. Finally, we obtain lower and upper bounds for the spectral norms of Kronecker and Hadamard products of Q and L matrices.