ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS

Abstract

In this paper, we compute the norms of circulant matrices with the complex Fibonacci and Lucas numbers. Moreover, we give golden ratio in complex Fibonacci numbers.In some scientific areas such as signal processing, coding theory and image processing, we often encounter circulant matrices. An n n  matrix C is called a circulant matrix if it is of the form 0 1 1 1 0 2 2 1 3 1 2 0 n n n n n n c c c c c c C c c c c c c                   or an n n matrix C is circulant if there exist0 1 1 , , ,nc c c such that the i, j entry of C is j i n mod c , where the rows and columns are numberedfrom 0 ton 1and kmodn means the number between 0ton 1that is congruent to kmodn. Thus, we denote thecirculant matrix C as C Circ c c c   0 1 1 , , ,n . Anycirculant matrix has many elegant properties. Some ofthem are [6,12]