Bounds of circulant matrices of norms with Pell and Pell Lucas numbers

In this study, firstly we define various norms, Kronecker product and Hadamard product basically and give relations among them. In addition we define briefly circulant matrix and give its some important properties. Then two of special integer sequences, Pell and Pell Lucas sequences are defined and describe some important identities and by using the variable we obtain a generalization of Pell, Pell Lucas sequences called k- Pell, k-Pell Lucas sequences. Then, we find the sum of the first n terms of Pell, Pell Lucas, k- Pell, k- Pell Lucas numbers. By using these formulas we find out upper and lower bounds of the spectral norm for the circulant matrix with Pell, Pell Lucas, k- Pell, k- Pell Lucas sequences. Besides this, we give upper and lower bounds of the spectral norm for the r-circulant matrix with Pell, Pell Lucas sequences.