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

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Received 14 December 2015; accepted 30 May 2016; published 2 June 2016

1. 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.

2. Preliminaries

The Fibonacci and Lucas sequences and are defined by the recurrence relations

and

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

and

If start from, then the Fibonacci and Lucas sequence are given by

The following sum formulas the Fibonacci and Lucas numbers are well known [11] [12] :

A matrix is called a r-Toeplitz matrix if it is of the form

(1)

Obviously, the r-Toeplitz matrix T is determined by parameter r and its first row elements, thus we denote. Especially, let, the matrix T is called a Toeplitz matrix.

A matrix is called a symmetric r-Toeplitz matrix if it is of the form

(2)

Obviously, the symmetric r-Toeplitz matrix T is determined by parameter r and its last row elements, thus we denote . Especially, let, the matrix T is called a Toeplitz matrix.

The Euclidean norm of the matrix A is defined as

The singular values of the matrix A is

where is an eigenvalue of and is conjugate transpose of matrix A. For a square matrix A, the square roots of the maximum eigenvalues of are called the spectral norm of A. The spectral norm of the matrix A is

The following inequality holds,

Define the maximum column lenght norm, and the maximum row lenght norm of any matrix A by

and

respectively. Let A, B and C be matrices. If then

[13] .

Theorem 1 [9] . Let be a Toeplitz matrix satisfying, then

where is the spectral norm and denotes the nth Fibonacci number.

Theorem 2 [9] . Let be a Toeplitz matrix satisfying, then

where is the spectral norm and denotes the nth Lucas number.

3. Result and Discussion

Theorem 3. Let be a r-Toeplitz matrix satisfying, where.

・

・

where is the spectral norm and denotes the nth Fibonacci number.

Proof. The matrix A is of the form

Then we have,

hence, when we obtain

that is

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

and

such that. Then

and

We have

when we also obtain

that is

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

and

such that. Then

and

We have

¢

Thus, the proof is completed.

Corollary 4. Let be a symmetric r-Toeplitz matrix, where r C, then

・

・

where is the spectral norm and 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. ¢

Theorem 5. Let be a r-Toeplitz matrix satisfying, where.

・

・

where is the spectral norm and denotes the nth Lucas number.

Proof. The matrix A is of the form

then we have

hence when we obtain

that is

On the other hand let matrices B and C be as

and

such that. Then

and

We have

when we also obtain

that is

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

and

such that. Then

and

We have

¢

Thus, the proof is completed.

Corollary 6. Let be a symmetric r-Toeplitz matrix, where, then

・

・

where is the spectral norm and 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. ¢

4. Numarical Examples

Example 7. Let be a r-Toeplitz matrix, in which denotes the Fibonacci number, where. From Table 1, it is easy to find that upper bounds for the spectral norm, of Theorem 3 are more sharper than Theorem 1 (see Table 1).

Table 1. Numerical results of,.

Table 2. Numerical results of,.

Example 8. Let be a r-Toeplitz matrix, in which denotes the Lucas number, where. From Table 2, it is easy to find that upper bounds for the spectral norm, of Theorem 5 are more sharper than Theorem 2, when n ≥ 2 (see Table 2).

NOTES

^{*}Corresponding author.

References

[1] Dubbs, A. and Edelman, A. (2014) Infinite Random Matrix Theory. Tridiagonal Bordered Toeplitz Matrices and the Moment Problem. arXiv:1502.04931v1.

[2] Erbas, C. and Tanik, M.M. (1995) Generating Solutions to the N-Queens Problems Using 2-Circulants. Mathematics Magazine, 68, 343-356.

http://dx.doi.org/10.2307/2690923

[3] Ngondiep, E., Serra-Capizzano, S. and Sesana, D. (2010) Spectral Features and Asymptotic Proporties for g-Circulant and g-Toeplitz Sequence. SIAM Journal on Matrix Analysis and Applications, 31, 1663-1687.

http://dx.doi.org/10.1137/090760209

[4] Szegö, G. (1958) Toeplitz Forms and Their Applications. University of California Press.

[5] Hudson, R.E., Reed, C.W., Chen, D. and Lorenzelli, F. (1998) Blind Beamforming on a Randomly Distributed sensor Array System. Journal on Selected Areas in Communications, 16, 1555-1567.

[6] Gray, R.M. and Davisson, L.D. (2005) An Introduction to Statistical Signal Processing. Cambridge University Press, London.

[7] Wei, Y., Cai, J. and Ng, M.K. (2004) Computing Moore-Penrose Inverses of Toeplitz Matrices by Newton’s Iteration. Mathematical and Computer Modelling, 40, 181-191.

http://dx.doi.org/10.1016/j.mcm.2003.09.036

[8] Chou, W.S., Du, B.S. and Shiue, P.J.-S. (2008) A Note on Circulant Transition Matrices in Markov Chains. Linear Algebra and Its Applications, 429, 1699-1704.

http://dx.doi.org/10.1016/j.laa.2008.05.004

[9] Akbulak, M. and Bozkurt, D. (2008) On the Norms of Toeplitz Matrices Involving Fibonacci and Lucas Numbers. Hacettepe Journal of Mathematics and Statistics, 37, 89-95.

[10] Shen, S. (2012) On the Norms of Toeplitz Matrices Involving k-Fibonacci and k-Lucas Numbers. International Journal of Contemporary Mathematical Sciences, 7, 363-368.

[11] Vajda, S. (1989) Fibonacci and Lucas Numbers and the Golden Section: Theory and Applications. Ellis Horwood Ltd.

[12] Koshy, T. (2001) Fibonacci and Lucas Numbers with Applications. A Wiley-Interscience Publication.

[13] Horn, R.A. and Johnson, C.R. (1991) Topics in Matrix Analysis. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511840371