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

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

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

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

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

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

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.