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Abstract and Applied Analysis
Volume 2015 (2015), Article ID 428146, 8 pages
http://dx.doi.org/10.1155/2015/428146
Research Article

Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

Department of Mathematics, Linyi University, Linyi, Shandong 276000, China

Received 25 July 2014; Accepted 16 September 2014

Academic Editor: Zidong Wang

Copyright © 2015 Wenai Xu and Zhaolin Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Circulant type matrices have played an important role in networks engineering. In this paper, firstly, some bounds for the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices and Lucas row skew first-minus-last right (RSFMLR) circulant matrices are given. Furthermore, the spectral norm of Hadamard product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is obtained. Finally, the Frobenius norm of Kronecker product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is presented.

1. Introduction

Circulant type matrices have been put on the firm basis with the work in [14] and so on. Circulant type matrices have significant applications in networks systems. In [5], some preliminary results on the dynamical behaviours of some specific nonmonotone Boolean automata networks which are called xor circulant networks were showed. In [6], the authors proposed a special class of the feedback delay network using circulant matrices. In [7], the impact of interior symmetries on the multiplicity of the eigenvalues of the Jacobian matrix at a fully synchronous equilibrium for the coupled cell systems associated with homogeneous networks was analyzed by Aguiar and Ruan, which was based on the circulant adjacency matrices of the networks induced by these interior symmetries. Exploiting the circulant structure of the channel matrices, the realistic near fast fading scenarios with circulant frequency selective channels were analysed by Eghbali et al. in [8]. The existence of doubly periodic travelling waves in cellular networks involving the discontinuous Heaviside step function by circulant matrix was studied by Wang and Cheng in [9].

The Fibonacci and Lucas sequences and are defined by the recurrence relations [10, 11]:If we start from , then Fibonacci and Lucas sequences are given byIn [10], their Binet forms are given by

The following sum formulations for the Fibonacci and Lucas numbers are well known [11]:Lately, some authors studied the problems of the norms of some special matrices [1121]. The author [11] found upper and lower bounds for the spectral norms of Toeplitz matrices such that and . In [13], the authors obtain upper and lower bounds for the spectral norms of matrices and , where and are -Fibonacci and -Lucas sequences, respectively, and they also give the bounds for the spectral norms of Kronecker and Hadamard products of these special matrices, respectively [14]. Solak and Bozkurt [16] have found out upper and lower bounds for the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices. Solak [1820] has defined and as circulant matrices, where and ; then he has given some bounds for the and matrices concerned with the spectral and Euclidean norms.

In this paper, we define two kinds of special matrices as follows.

A Fibonacci row skew first-minus-last right (RSFMLR) circulant matrix is defined as a square matrix of the form

A Lucas row skew first-minus-last right (RSFMLR) circulant matrix is defined as a square matrix of the formObviously, the RSFMLR circulant matrix is determined by its first row, and RSFMLR circulant matrix is a circulant matrix [22].

We define as the basic RSFMLR circulant matrix; that is,It is easily verified that has no repeated roots in its splitting field and is both the minimal polynomial and the characteristic polynomial of the matrix . In addition, is nonderogatory and satisfies and .

As we all know, letting be a RSFMLR circulant matrix with the first row , it is clear thatThus, is a RSFMLR circulant matrix if and only if for some polynomial . The polynomial will be called the representer of the RSFMLR circulant matrix . By (11), it is clear that is a RSFMLR circulant matrix if and only if commutes with ; that is, .

In addition to the algebraic properties that can be easily derived from the representation (11), we mention that RSFMLR circulant matrices have very nice structure. The product of two RSFMLR circulant matrices is a RSFMLR circulant matrix and is a RSFMLR circulant matrix too.

Let be an matrix. The Euclidean (or Frobenius) norm, the spectral norm, the maximum column sum matrix norm, and the maximum row sum matrix norm of the matrix are, respectively [11],where denotes the conjugate transpose of . The following inequality holds:Let and be matrices. The Hadamard product of and is defined by . If is any norm on matrices, then [18, 23]Kronecker product of and is given to be [18]Then [18]Let be an matrix with eigenvalues , . The spread of is defined as [24, 25]An upper bound for the spread due to Mirsky [24] states thatwhere denotes the Frobenius norm of and is the trace of .

2. Norms and Spread of Fibonacci RSFMLR Circulant Matrices

Theorem 1. Let be a Fibonacci RSFMLR circulant matrix, where denote Fibonacci numbers given by (1); then two kinds of norms of are given by

Proof. The matrix is of the form (8), by (14), (15); then we haveSince the Fibonacci sequences are defined by the recurrence relations (1), then we obtainTo sum up, we can get Thenwhich completes the proof.

Theorem 2. Let be a Fibonacci RSFMLR circulant matrix, where denote Fibonacci numbers given by (1); thenwhere

Proof. Since and given by (1), the matrix is of the form We know that from equivalent norms. By (5), we can getThenwhereWe haveOn the other hand, suppose thatThenWe can getFurthermore,We obtainThe other result is obtained as follows:which completes the proof.

Theorem 3. Let be a Fibonacci RSFMLR circulant matrix, where denote Fibonacci numbers given by (1); then the bound for the spread of iswhere

Proof. The trace of is . By Theorem 2 and inequation (21), we havewhereWe can getwherewhich completes the proof.

3. Norms and Spread of Lucas RSFMLR Circulant Matrices

Theorem 4. Let be a Lucas RSFMLR circulant matrix, where denote Lucas numbers given by (2); then two kinds of norms of are given by

Proof. The matrix is of the form (9), by (14), (15); then we getSince the Lucas sequences are defined by the recurrence relations (2), then we obtainTo sum up, we can getThenwhich completes the proof.

Theorem 5. Let be a Lucas RSFMLR circulant matrix, where denote Lucas numbers given by (2); thenwhere

Proof. Since and , the matrix is of the form We know that from equivalent norms. By (6), we can getThenWe havewhereOn the other hand, supposing thatthenWe obtainWe haveWe getThe other result is obtained as follows:which completes the proof.

Theorem 6. Let be a Lucas RSFMLR circulant matrix, where denote Lucas numbers given by (2); then where

Proof. The trace of is . By Theorem 5 and by inequation (21), we havewhereWe obtainwherewhich completes the proof.

Corollary 7. Let be a Fibonacci RSFMLR circulant matrix and let be a Lucas RSFMLR circulant matrix, where and denote Fibonacci numbers and Lucas numbers, respectively; then the spectral norm of Hadamard product of and satisfies the following inequality:

Proof. The proof is trivial by Theorems 2 and 5; we obtainBy inequation (17), we havewhich completes the proof.

Corollary 8. Let be a Fibonacci RSFMLR circulant matrix and let be a Lucas RSFMLR circulant matrix, where and denote Fibonacci numbers and Lucas numbers, respectively; then the Frobenius norm of Kronecker product of and iswhere

Proof. Since the proof is trivial by Theorems 2 and 5, we obtainBy (19), thenwherewhich completes the proof.

4. Conclusion

In this study, we define matrices of the following forms: let be a Fibonacci RSFMLR circulant matrix and let be a Lucas RSFMLR circulant matrix. Firstly, we get lower and upper bounds for the spectral norms of these matrices. Upper bounds for the spread of the matrix and the matrix are given. Afterwards, we obtain some corollaries related to norms of Hadamard and Kronecker products of these matrices. Based on the existing problems in [2628], we will explore solving these problems by circulant matrices technology.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and the AMEP of Linyi University, China.

References

  1. P. J. Davis, Circulant Matrices, John Wiley & Sons, New York, NY, USA, 1979. View at MathSciNet
  2. Z. L. Jiang and Z. X. Zhou, Circulant Matrices, Chengdu Technology University Publishing Company, Chengdu, China, 1999.
  3. Z. Jiang, “On the minimal polynomials and the inverses of multilevel scaled factor circulant matrices,” Abstract and Applied Analysis, vol. 2014, Article ID 521643, 10 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  4. X. Jiang and K. Hong, “Exact determinants of some special circulant matrices involving four kinds of famous numbers,” Abstract and Applied Analysis, vol. 2014, Article ID 273680, 12 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  5. M. Noual, D. Regnault, and S. Sené, “About non-monotony in Boolean automata networks,” Theoretical Computer Science, vol. 504, pp. 12–25, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. D. Rocchesso and J. O. Smith, “Circulant and elliptic feedback delay networks for artificial reverberation,” IEEE Transactions on Speech and Audio Processing, vol. 5, no. 1, pp. 51–63, 1997. View at Publisher · View at Google Scholar · View at Scopus
  7. M. A. Aguiar and H. Ruan, “Interior symmetries and multiple eigenvalues for homogeneous networks,” SIAM Journal on Applied Dynamical Systems, vol. 11, no. 4, pp. 1231–1269, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. H. Eghbali, S. Muhaidat, S. A. Hejazi, and Y. Ding, “Relay selection strategies for single-carrier frequency-domain equalization multi-relay cooperative networks,” IEEE Transactions on Wireless Communications, vol. 12, no. 5, pp. 2034–2045, 2013. View at Publisher · View at Google Scholar · View at Scopus
  9. G.-Q. Wang and S. S. Cheng, “6-periodic travelling waves in an artificial neural network with bang-bang control,” Journal of Difference Equations and Applications, vol. 18, no. 2, pp. 261–304, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. E. G. Kocer, N. Tuglu, and A. Stakhov, “On the m-extension of the Fibonacci and Lucas p-numbers,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1890–1906, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. M. Akbulak and D. Bozkurt, “On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers,” Hacettepe Journal of Mathematics and Statistics, vol. 37, no. 2, pp. 89–95, 2008. View at Google Scholar · View at MathSciNet · View at Scopus
  12. R. Mathias, “The spectral norm of a nonnegative matrix,” Linear Algebra and Its Applications, vol. 139, pp. 269–284, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Q. Shen and J. M. Cen, “On the spectral norms of r-circulant matrices with the k-fibonacci and k-lucas numbers,” International Journal of Contemporary Mathematical Sciences, vol. 5, no. 12, pp. 569–578, 2010. View at Google Scholar
  14. S. Q. Shen and J. M. Cen, “On the norms of circulant matrices with the (k, h)-fibonacci and (k, h)-lucas numbers,” International Journal of Contemporary Mathematical Sciences, vol. 6, no. 18, pp. 887–894, 2011. View at Google Scholar · View at MathSciNet
  15. S. Solak and D. Bozkurt, “Some bounds on lp matrix and lp operator norms of almost circulant, CAUchy-Toeplitz and CAUchy-Hankel matrices,” Mathematical & Computational Applications, vol. 7, no. 3, pp. 211–218, 2002. View at Google Scholar · View at MathSciNet
  16. S. Solak and D. Bozkurt, “On the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 231–238, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. S. Solak and D. Bozkurt, “A note on bound for norms of Cauchy-Hankel matrices,” Numerical Linear Algebra with Applications, vol. 10, no. 4, pp. 377–382, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. S. Solak, “On the norms of circulant matrices with the Fibonacci and Lucas numbers,” Applied Mathematics and Computation, vol. 160, no. 1, pp. 125–132, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. S. Solak, “Erratum to “on the norms of circulant matrices with the Fibonacci and Lucas numbers”,” Applied Mathematics and Computation, vol. 160, no. 1, pp. 125–132, 2005. View at Google Scholar
  20. S. Solak, “Erratum to “On the norms of circulant matrices with the Fibonacci and Lucas numbers” [Appl. Math. Comput. 160 (2005) 125–132],” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1855–1856, 2007. View at Publisher · View at Google Scholar
  21. A. Yalc'iner, “Spectral norms of some special circulant matrices,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 35, pp. 1733–1738, 2008. View at Google Scholar
  22. D. Chillag, “Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes,” Linear Algebra and Its Applications, vol. 218, pp. 147–183, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  23. G. Visick, “A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product,” Linear Algebra and Its Applications, vol. 304, no. 1–3, pp. 45–68, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  24. L. Mirsky, “The spread of a matrix,” Mathematika, vol. 3, pp. 127–130, 1956. View at Google Scholar · View at MathSciNet
  25. R. Sharma and R. Kumar, “Remark on upper bounds for the spread of a matrix,” Linear Algebra and its Applications, vol. 438, no. 11, pp. 4359–4362, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. J. Hu, Z. Wang, and H. Gao, “Recursive filtering with random parameter matrices, multiple fading measurements and correlated noises,” Automatica, vol. 49, no. 11, pp. 3440–3448, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. D. Ding, Z. Wang, B. Shen, and H. Shu, “State-saturated H filtering with randomly occurring nonlinearities and packet dropouts: the finite-horizon case,” International Journal of Robust and Nonlinear Control, vol. 23, no. 16, pp. 1803–1821, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. B. Shen, Z. Wang, D. Ding, and H. Shu, “H state estimation for complex networks with uncertain inner coupling and incomplete measurements,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 12, pp. 2027–2037, 2013. View at Publisher · View at Google Scholar · View at Scopus