On the Norms of r-Toeplitz Matrices Involving Fibonacci and Lucas Numbers

Let us define     r ij A T a = to be a × n n r -Toeplitz matrix. The entries in the first row of     r ij A T a = are ij i j a F − = or ij i j a L − = where Fn and Ln denote the usual Fibonacci and Lucas numbers, respectively. We obtained some bounds for the spectral norm of these matrices.


Introduction
Toeplitz matrices arise in many different theoretical and applicative fields, in the mathematical modeling of all the problems where some sort of shift invariance occurs in terms of space or of time.As in computation of spline functions, time series analysis, signal and image processing, queueing theory, polynomial and power series computations and many other areas, typical problems modelled by Toeplitz matrices are the numerical solution of certain differential and integral equations [1]- [5].
Lots of article have been written so far, which concern estimates for spectral norms of Toeplitz matrices, which have connections with signal and image processing, time series analysis and many other problems [6]- [8].Akbulak and Bozkurt found lower and upper bounds for the spectral norms of Toeplitz matrices with classical Fibonacci and Lucas numbers entries in [9].Shen gave upper and lower bounds for the spectral norms of Toeplitz matrices with k-Fibonacci and k-Lucas numbers entries in [10].
In this paper, we derive expressions of spectral norms for r-Toeplitz matrices.We explain some preliminaries and well-known results.We thicken the identities of estimations for spectral norms of r-Toeplitz matrices.

Preliminaries
The Fibonacci and Lucas sequences n F and n L are defined by the recurrence relations The rule can be used to extend the sequence backwards.Hence ( ) The following sum formulas the Fibonacci and Lucas numbers are well known [11] [12]: Obviously, the r-Toeplitz matrix T is determined by parameter r and its first row elements , thus we denote ( ) Obviously, the symmetric r-Toeplitz matrix T is determined by parameter r and its last row elements The Euclidean norm of the matrix A is defined as The singular values of the matrix A is ( ) λ is an eigenvalue of * A A and * A is conjugate transpose of matrix A. For a square matrix A, the square roots of the maximum eigenvalues of * A A are called the spectral norm of A. The spectral norm of the matrix A is ( ) The following inequality holds, 2 1 .
Define the maximum column lenght norm 1 c , and the maximum row lenght norm 1 r of any matrix A by ( ) Theorem 1 [9].
. is the spectral norm and n F denotes the nth Fibonacci number.Theorem 2 [9].
. is the spectral norm and n L denotes the nth Lucas number.

Result and Discussion
. is the spectral norm and n F denotes the nth Fibonacci number.Proof.The matrix A is of the form On the other hand, let the matrices B and C as We have ( ) On the other hand, let the matrices B and C as .
be a symmetric r-Toeplitz matrix, where r C, then . is the spectral norm and n F denotes the nth Fibonacci number.Proof.Owing to the fact that the sum of all elements squares are equal in matrices ( 1) and ( 2), the proof is concluded analogously in the proof of previous theorem.
. is the spectral norm and n L denotes the nth Lucas number.Proof.The matrix A is of the form hence when 1 r ≥ we obtain ( ) On the other hand let matrices B and C be as when 1 r < we also obtain ( ) On the other hand, let matrices B and C be as . is the spectral norm and n L denotes the nth Lucas number.
Proof.Owing to the fact that the sum of all elements squares are equal in matrices ( 1) and ( 2), the proof is concluded analogously in the proof of previous theorem.    2, it is easy to find that upper bounds for the spectral norm, of Theo- rem 5 are more sharper than Theorem 2, when n ≥ 2 (see Table 2).

Numarical Examples
number, where r ∈  .From Table then the Fibonacci and Lucas sequence are given by