^{1}

^{2}

^{*}

Let us define
to be a
*r*-Toeplitz matrix. The entries in the first row of
are
or
;where
F_{n} and
L_{n} denote the usual Fibonacci and Lucas numbers, respectively. We obtained some bounds for the spectral norm of these matrices.

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 [

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 [

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.

The Fibonacci and Lucas sequences

and

The rule can be used to extend the sequence backwards. Hence

and

If start from

The following sum formulas the Fibonacci and Lucas numbers are well known [

A matrix

Obviously, the r-Toeplitz matrix T is determined by parameter r and its first row elements

A matrix

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

where

The following inequality holds,

Define the maximum column lenght norm

and

respectively. Let A, B and C be

Theorem 1 [

where

Theorem 2 [

where

Theorem 3. Let

・

・

where

Proof. The matrix A is of the form

Then we have,

hence, when

that is

On the other hand, let the matrices B and C as

and

such that

and

We have

when

that is

On the other hand, let the matrices B and C as

and

such that

and

We have

¢

Thus, the proof is completed.

Corollary 4. Let

・

・

where

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. ¢

Theorem 5. Let

・

・

where

Proof. The matrix A is of the form

then we have

hence when

that is

On the other hand let matrices B and C be as

and

such that

and

We have

when

that is

On the other hand, let matrices B and C be as

and

such that

and

We have

¢

Thus, the proof is completed.

Corollary 6. Let

・

・

where

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. ¢

Example 7. Let

n | Theorem 1 | Theorem 3 |
---|---|---|

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

・・・ | ・・・ | ・・・ |

n |

n | Theorem 2 | Theorem 5 |
---|---|---|

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

・・・ | ・・・ | ・・・ |

n |

Example 8. Let

Hasan Gökbaş,Ramazan Türkmen, (2016) On the Norms of r-Toeplitz Matrices Involving Fibonacci and Lucas Numbers. Advances in Linear Algebra & Matrix Theory,06,31-39. doi: 10.4236/alamt.2016.62005