Abstract and Applied Analysis

Volume 2014, Article ID 381829, 9 pages

http://dx.doi.org/10.1155/2014/381829

## Determinants, Norms, and the Spread of Circulant Matrices with Tribonacci and Generalized Lucas Numbers

^{1}Department of Mathematics, Linyi University, Linyi, Shandong 276000, China^{2}Department of Mathematics, Shandong Normal University, Ji’nan, Shandong 250014, China

Received 28 March 2014; Accepted 26 April 2014; Published 11 May 2014

Academic Editor: Tongxing Li

Copyright © 2014 Juan Li et al. 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 matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degree *n*, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

#### 1. Introduction

Circulant matrices have important applications in solving various partial differential equations. By the radial properties of the fundamental solution and radial symmetric of the solution domain, Chen et al. [1] showed the circulant or block circulant features of the coefficient matrices for problems under pure Dirichlet or Neumann boundary condition. Lei and Sun [2] proposed the preconditioned CGNR (PCGNR) method with a circulant preconditioner to solve such Toeplitz-like systems. Using circulant matrix, Karasozen and Simsek [3] considered periodic boundary conditions such that no additional boundary terms will appear after semidiscretization. In [4], a semicirculant preconditioner applied to a problem, subject to Dirichlet boundary conditions at the inflow boundaries, was examined. In [5], the resulting dense linear system exhibits so much structure that it can be solved very efficiently by a circulant preconditioned conjugate gradient method. A method was described for obtaining finite difference approximation solutions of multidimensional partial differential equations satisfying boundary conditions specified on irregularly shaped boundaries by using circulant matrices and fast Fourier transform (FFT) convolutions in [6]. Brockett and Willems [7] showed how the important problems of linear system theory can be solved concisely for a particular class of linear systems, namely, block circulant systems, by exploiting the algebraic structure. The main theory of circulant dynamics considered in [8] is about circulant matrix.

Circulant matrices also play an important role in solving ordinary differential equations. By using a Strang-type block circulant preconditioner, Zhang et al. [9] speeded up the convergent rate of boundary-value methods. Delgado et al. [10] developed some techniques to obtain global hyperbolicity for a certain class of endomorphisms of with ; this kind of endomorphisms was obtained from vectorial difference equations where the mapping defining these equations satisfies a circulant matrix condition. In [11], nonsymmetric, large, and sparse linear systems were solved by using the generalized minimal residual (GMRES) method; a circulant block preconditioner was proposed to speed up the convergence rate of the GMRES method. Wilde [12] developed a theory for the solution of ordinary and partial differential equations whose structure involves the algebra of circulants. He showed how the algebra of circulants relates to the study of the harmonic oscillator, the Cauchy-Riemann equations, Laplace's equation, the Lorentz transformation, and the wave equation. And he used circulants to suggest natural generalizations of these equations to higher dimensions.

Circulant matrices have important applications in various disciplines including image processing [13–15], communications, signal processing [16], encoding, solving Toeplitz matrix problems, preconditioner, and solving least squares problems. They have been put on firm basis with the work of Davis [17] and Jiang and Zhou [18].

Some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [17, 18]. Unfortunately, the computational complexity of these algorithms is exorbitant with the order of matrix increasing. However, some authors gave the explicit determinants and inverses of circulant and skew circulant involving some famous numbers. For example, Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers [19]. Lind presented the determinants of circulant and skew circulant involving Fibonacci numbers [20]. Lin [21] gave the determinant of the Fibonacci-Lucas quasi-cyclic matrices. Shen considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices [22]. Gao et al. [23] gave explicit determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas numbers. Jiang et al. [24, 25] considered the skew circulant and skew left circulant matrices with the -Fibonacci numbers and the -Lucas numbers and discussed the invertibility of these matrices and presented their determinant and inverse matrix by constructing the transformation matrices, respectively.

Recently, there are several papers on the norms of some special matrices. Solak [26] established the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries. Ipek [27] investigated an improved estimation for spectral norms of these matrices. Shen and Cen [28] gave upper and lower bounds for the spectral norms of -circulant matrices in the forms of and , and they also obtained some bounds for the spectral norms of Kronecker and Hadamard products of matrix and matrix . Akbulak and Bozkurt [29] found upper and lower bounds for the spectral norms of Toeplitz matrices such that and . The convergence in probability and the convergence in distribution of the spectral norm of scaled Toeplitz, circulant, reverse circulant, symmetric circulant, and a class of -circulant matrices are discussed in [30].

Beginning with Mirsky [31], several authors [32–34] have obtained bounds for the spread of a matrix.

In this paper, by using the inverse factorization of polynomial of degree , the explicit determinants of the circulant and left circulant matrix involving Tribonacci numbers and generalized Lucas numbers are expressed by utilizing only Tribonacci numbers and generalized Lucas numbers. Furthermore, the norms and some upper and lower bounds for the spread of these matrices are given, respectively.

The Tribonacci sequence and the generalized Lucas sequence are defined by a third-order recurrence [35–37]: with the initial conditions , , and , , .

A few values of the sequences are given by the following table:

Note that are the roots of the characteristic equation . Then, the Binet formulae of the sequences and are where , .

The relation between the roots and coefficient in the characteristic equations is

Lemma 1. *Several formulae concerning these sequences are listed as follows:
*

*Proof. *Firstly, we can find formula (7) in [37]. Secondly, we give the computation about the sum of the first numbers of those sequences.

According to the recurrence relations (1) and (4) and Binet formula of , we can get
The assertions about the representation can be proved in the same way.

Finally, the quadratic sum of generalized Lucas sequences can be obtained as follows:

Hence, the proof is completed.

*Definition 2 (see [17, 18]). *A circulant matrix , denoted by , is a matrix of the form

*Definition 3 (see [17, 18]). *A left circulant matrix , denoted by , is a matrix of the form

*Let and . By explicit computation, we find
where .*

*Definition 4 (see [29]). *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,
where denotes the conjugate transpose of .

*Definition 5 (see [32]). *Let be an matrix with eigenvalues , . The spread of is defined as

*An upper bound for the spread due to Mirsky [31] states that
where denotes the Frobenius norm of and is the trace of .*

*Lemma 6 (see [34]). If is an matrix, then(i)if is real and normal, then ;(ii)if is Hermitian, then .*

*Lemma 7 (see [30]). If is an real symmetric or normal matrix, then we have , where () are the eigenvalues of .*

*Lemma 8 (see [17, 18]). Let () be the roots of the equation . If , then the eigenvalues and determinant of are and , respectively.*

*Lemma 9 (see [20]). Let () satisfy the equation ; then , .*

*Lemma 10 (see [38]). Let , , and . If , then the eigenvalues of are given by
and , is the largest integer less than or equal to . Note that, if is even, then .*

*2. Determinant, Norms, and the Spread of Circulant and Left Circulant Matrices with Tribonacci Numbers*

*2. Determinant, Norms, and the Spread of Circulant and Left Circulant Matrices with Tribonacci Numbers*

*Theorem 11. Let . Then the determinant of is
where
*

*Proof. *According to Lemma 8 and the Binet form of , we obtain that the eigenvalues of are
where and () are the roots of . From (4), we have
where () are the roots of equation . Applying Lemma 9, we have
where

*Theorem 12. Let . Then three kinds of norms of are given by
*

*Proof. *On the basis of the definitions of norms and (5) in Lemma 1, we have .

According to the definition of norms and (7) in Lemma 1, we know that
hence, the Frobenius norm of is

*Theorem 13. Let ; then the spectral norm of is
*

*Proof. *The modules of the eigenvalues of satisfy
which implies that is an eigenvalue of and . Hence, by Lemma 7 and equality (5) in Lemma 1, we have .

*Theorem 14. Let . Then the bounds for the spread of are
*

*Proof. *The trace of is and . Since is a real and normal matrix, by using Lemma 6, we can get .

Beside that, by Theorem 12, we have
by (16), we obtain

*Theorem 15. Let . Then the determinant of is
*

*Proof. *Since
the result can be derived from Theorem 11 and relation (13).

*Theorem 16. Let ; then
*

*Proof. *By the definition of norms and formula (5) in Lemma 1, we know that .

According to Definition 4 and (7) in Lemma 1, we have
thus, the Frobenius norm of is

*Theorem 17. Let ; then the spectral norm of is
*

*Proof. *Obviously, the modules of the first eigenvalues of are
and by Lemma 10. Since
we have . Beside that, if is even, then
In other words, for any , we have , and is an eigenvalue of . So, . Since is a real symmetric matrix, we can get by Lemma 7 and (5) in Lemma 1.

*Theorem 18. Let ; then the bounds for the spread of are
where
*

*Proof. *It follows from the elements in that ; since is a Hermitian matrix, so .

If is odd, the trace of is . By using Theorem 16, we know that

If is even, the trace of is
by using Theorem 16, we have
According to (16), the proof is completed.

*3. Determinant, Norms, and the Spread of Circulant and Left Circulant Matrices with Generalized Lucas Numbers*

*3. Determinant, Norms, and the Spread of Circulant and Left Circulant Matrices with Generalized Lucas Numbers**Theorem 19. Let . Then the determinant of is
where
*

*Proof. *By Lemma 8 and the Binet form of , the eigenvalues of are
according to (4), we have
where () are the roots of equation .

According to Lemma 9, we have
where

*Theorem 20. Let ; then the norms of are
*

*Proof. *According to the definition of norms and formula (6) in Lemma 1, we obtain .

According to the definition of norms and (8) in Lemma 1, we can get
thus, the spectral norm of is

*Theorem 21. Let ; then the spectral norm of is
*

*Proof. *The modules of the eigenvalues of satisfy
which means that is an eigenvalue of , so . Hence, the spectral norm of is by Lemma 7 and formula (6) in Lemma 1.

*Theorem 22. Let ; then the bounds for the spread of are
*

*Proof. *The trace of is and . Since is a real normal matrix, by Lemma 6, we can get
Beside that, by Theorem 20, we have
By (16), the proof is completed.

*Theorem 23. Let . Then
*

*Proof. *The conclusion can be proved by Theorem 19 and relation (13).

*Theorem 24. Let ; then the norms of are
*

*Proof. *According to the definition of norm and formula (6) in Lemma 1, we have .

According to the definition of norm and (8) in Lemma 1, we can get
thus, the Frobenius norm of is .

*Theorem 25. Let ; then the spectral norm of is
*

*Proof. *Obviously, the modules of the first eigenvalues of are
and by Lemma 10. Since
we have . Beside that, if is even, then
In other words, for any , we have , and is an eigenvalue of . So . Since is a real symmetric matrix, we can get by Lemma 7 and (6) in Lemma 1.

*Theorem 26. Let ; then the bounds for the spread of are
*

*Proof. *From the elements in , we know that ; since is a Hermitian matrix, so .

If is odd, the trace of is ; by using Theorem 24, we have

If is even,
by using Theorem 24, we have
According to (16), the conclusions are obtained.

*4. Conclusion*

*4. Conclusion**The related problems of circulant matrix and some famous numbers are studied in this paper. We not only study basic properties of circulant matrix or famous numbers, respectively, but also explore the explicit determinant and the four kinds of norms and give the upper and lower bounds for the spread of circulant matrix and left circulant matrix involving Tribonacci numbers and generalized Lucas numbers. If we combine famous numbers with circulant matrix and left circulant matrix, a lot of good results would be obtained, and we wish the results could be useful in solving some differential equations.*

*Conflict of Interests*

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

*Acknowledgments*

*Acknowledgments**The research was supported by the Development Project of Science and Technology of Shandong Province (Grant no. 2012GGX10115) and NSFC (Grant no. 11301251) and the AMEP of Linyi University, China.*

*References*

*References*

- W. Chen, J. Lin, and C. S. Chen, “The method of fundamental solutions for solving exterior axisymmetric Helmholtz problems with high wave-number,”
*Advances in Applied Mathematics and Mechanics*, vol. 5, no. 4, pp. 477–493, 2013. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-L. Lei and H.-W. Sun, “A circulant preconditioner for fractional diffusion equations,”
*Journal of Computational Physics*, vol. 242, pp. 715–725, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - B. Karasozen and G. Simsek, “Energy preserving integration of bi-Hamiltonian partial differential equations,”
*Applied Mathematics Letters*, vol. 26, no. 12, pp. 1125–1133, 2013. View at Google Scholar - K. Otto, “Analysis of preconditioners for hyperbolic partial differential equations,”
*SIAM Journal on Numerical Analysis*, vol. 33, no. 6, pp. 2131–2165, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - E. W. Sachs and A. K. Strauss, “Efficient solution of a partial integro-differential equation in finance,”
*Applied Numerical Mathematics*, vol. 58, no. 11, pp. 1687–1703, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. E. Gilmour, “Circulant matrix methods for the numerical solution of partial differential equations by FFT convolutions,”
*Applied Mathematical Modelling*, vol. 12, no. 1, pp. 44–50, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - R. W. Brockett and J. L. Willems, “Discretized partial differential equations: examples of control systems defined on modules,”
*Automatica*, vol. 10, no. 5, pp. 507–515, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. E. Cohn and D. P. Dee, “Observability of discretized partial differential equations,”
*SIAM Journal on Numerical Analysis*, vol. 25, no. 3, pp. 586–617, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Zhang, H. Chen, and L. Wang, “Strang-type preconditioners applied to ordinary and neutral differential-algebraic equations,”
*Numerical Linear Algebra with Applications*, vol. 18, no. 5, pp. 843–855, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Delgado, N. Romero, A. Rovella, and F. Vilamajó, “Bounded solutions of quadratic circulant difference equations,”
*Journal of Difference Equations and Applications*, vol. 11, no. 10, pp. 897–907, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X.-Q. Jin, V.-K. Sin, and L.-L. Song, “Circulant-block preconditioners for solving ordinary differential equations,”
*Applied Mathematics and Computation*, vol. 140, no. 2-3, pp. 409–418, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. C. Wilde, “Differential equations involving circulant matrices,”
*The Rocky Mountain Journal of Mathematics*, vol. 13, no. 1, pp. 1–13, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-J. Wittsack, A. M. Wohlschläger, E. K. Ritzl et al., “CT-perfusion imaging of the human brain: advanced deconvolution analysis using circulant singular value decomposition,”
*Computerized Medical Imaging and Graphics*, vol. 32, no. 1, pp. 67–77, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. F. Henriques, R. Caseiro, P. Martins, and J. Batista, “Exploiting the circulant structure of tracking-by-detection with kernels,” in
*Proceedings of the European Conference on Computer Vision (ECCV '12)*, pp. 1–14, 2012. - X. Huang, G. Ye, and K.-W. Wong, “Chaotic image encryption algorithm based on circulant operation,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 384067, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Andrecut, “Applications of left circulant matrices in signal and image processing,”
*Modern Physics Letters B*, vol. 22, no. 4, pp. 231–241, 2008. View at Publisher · View at Google Scholar · View at Scopus - P. J. Davis,
*Circulant Matrices*, John Wiley & Sons, New York, NY, USA, 1979. View at MathSciNet - Z. L. Jiang and Z. X. Zhou,
*Circulant Matrices*, Chengdu Technology University, Chengdu, China, 1999. - D. V. Jaiswal, “On determinants involving generalized Fibonacci numbers,”
*The Fibonacci Quarterly*, vol. 7, pp. 319–330, 1969. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. A. Lind, “A Fibonacci circulant,”
*The Fibonacci Quarterly*, vol. 8, no. 5, pp. 449–455, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Lin, “Fibonacci-Lucas quasi-cyclic matrices,”
*The Fibonacci Quarterly*, vol. 40, no. 3, pp. 280–286, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S.-Q. Shen, J.-M. Cen, and Y. Hao, “On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers,”
*Applied Mathematics and Computation*, vol. 217, no. 23, pp. 9790–9797, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Gao, Z. L. Jiang, and Y. P. Gong, “On the determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas Numbers,”
*WSEAS Transactions on Mathematics*, vol. 12, no. 4, pp. 472–481, 2013. View at Google Scholar - X. Y. Jiang, Y. Gao, and Z. L. Jiang, “Determinants and inverses of skew and skew left circulant matrices involving the k-Fibonacci numbers in communications-I,”
*Far East Journal of Mathematical Sciences*, vol. 76, no. 1, pp. 123–137, 2013. View at Google Scholar - X. Y. Jiang, Y. Gao, and Z. L. Jiang, “Determinants and inverses of skew and skew left circulant matrices involving the k-Lucas numbers in communications-II,”
*Far East Journal of Mathematical Sciences*, vol. 78, no. 1, pp. 1–17, 2013. View at Google Scholar - 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 Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Ipek, “On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 6011–6012, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Shen and J. Cen, “On the bounds for the norms of
*r*-circulant matrices with the Fibonacci and Lucas numbers,”*Applied Mathematics and Computation*, vol. 216, no. 10, pp. 2891–2897, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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 Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Bose, R. S. Hazra, and K. Saha, “Spectral norm of circulant-type matrices,”
*Journal of Theoretical Probability*, vol. 24, no. 2, pp. 479–516, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - L. Mirsky, “The spread of a matrix,”
*Mathematika*, vol. 3, pp. 127–130, 1956. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 Zentralblatt MATH · View at MathSciNet - J. Wu, P. Zhang, and W. Liao, “Upper bounds for the spread of a matrix,”
*Linear Algebra and its Applications*, vol. 437, no. 11, pp. 2813–2822, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. R. Johnson, R. Kumar, and H. Wolkowicz, “Lower bounds for the spread of a matrix,”
*Linear Algebra and Its Applications*, vol. 71, pp. 161–173, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Elia, “Derived sequences, the Tribonacci recurrence and cubic forms,”
*The Fibonacci Quarterly*, vol. 39, no. 2, pp. 107–115, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - B. Balof, “Restricted tilings and bijections,”
*Journal of Integer Sequences*, vol. 15, Article ID 12.2.3, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Rabinowitz, “Algorithmic manipulation of third-order linear recurrences,”
*The Fibonacci Quarterly*, vol. 34, no. 5, pp. 447–464, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Bose and J. Mitra, “Limiting spectral distribution of a special circulant,”
*Statistics and Probability Letters*, vol. 60, no. 1, pp. 111–120, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus

*
*