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 exist
0 1 1 , , ,
n
c c c 
such that the i, j entry of C is
 j i n mod c 
, where the rows and columns are numbered
from 0 to
n 1
and kmodn means the number between 0
to
n 1
that is congruent to kmodn. Thus, we denote the
circulant matrix C as C Circ c c c   0 1 1 , , ,
n 
. Any
circulant matrix has many elegant properties. Some of
them are [6,12]

Keywords

• Circulant matrix
• Complex Fibonacci numbers
• Matrix norm.

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