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

Abstract

Abstract In this paper, we present new upper and lower bounds for the spectral norms of the r-circulant matrices Q=Cr((ba)ξ(1)2q0,(ba)ξ(2)2q1,(ba)ξ(3)2q2,…,(ba)ξ(n)2qn−1) $Q=C_{r} ( (\frac{b}{a} )^{\frac{\xi (1)}{2}}q_{0}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}q_{1}, (\frac {b}{a} )^{\frac{\xi(3)}{2}}q_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n)}{2}}q_{n-1} )$ and L=Cr((ba)ξ(0)2l0,(ba)ξ(1)2l1,(ba)ξ(2)2l2,…,(ba)ξ(n−1)2ln−1) $L=C_{r} ( (\frac {b}{a} )^{\frac{\xi(0)}{2}}l_{0}, (\frac{b}{a} )^{\frac{\xi (1)}{2}}l_{1}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}l_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n-1)}{2}}l_{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.

Keywords

• biperiodic Fibonacci number
• biperiodic Lucas number
• r-circulant matrix
• norm