On Jacobsthal and Jacobsthal-Lucas Circulant Type Matrices

Abstract

Circulant type matrices have become an important tool in solving fractional order differential equations. In this paper, we consider the circulant and left circulant and -circulant matrices with the Jacobsthal and Jacobsthal-Lucas numbers. First, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix. Furthermore, the invertibility of the left circulant and -circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and -circulant matrices by utilizing the relation between left circulant, -circulant matrices, and circulant matrix, respectively.

1. Introduction

Circulant type matrices possess interesting properties, which have been exploited to obtain the transient solution in a closed form for fractional order differential equations. In [1], Qu et al. derived the discretized linear system from fractional advection-diffusion equations (FADEs) with constant coefficients and introduced the background of circulant and skew-circulant splitting (CSCS) iteration for solving the discretized linear system. In [2], some Routh-Hurwitz stability conditions are generalized to the fractional order case. Ahmed et al. considered the 1-system CML (10) in [2]. They selected a circulant matrix, which reads a tridiagonal matrix. Ahmed et al. used coupled map lattices (CML) as an alternative approach to include spatial effects in fractional order systems (FOS). Consider the 1-system CML (10) in [3]. They claimed that the system is stable if all the eigenvalues of the circulant matrix satisfy (2) in [3]. Lei and Sun [4] proposed the preconditioned CGNR (PCGNR) method with a circulant preconditioner to solve such Toeplitz-like systems. Kloeden et al. adopted the multilevel Monte Carlo method introduced by M. Giles to SDEs with additive fractional noise. They adopted the simplest approximation schemes for (1) in [5] with the Euler method, which reads (5) in [5]. They exploited that the covariance matrix of the increments can be embedded in a circulant matrix. The total loops can be done by fast Fourier transformation, which leads to a total computational cost of , where is the order of the circulant matrix.

Jacobsthal and Jacobsthal-Lucas circulant type matrices are circulant type matrices with Jacobsthal and Jacobsthal-Lucas numbers. Circulant type matrices include the circulant and left circulant and -circulant matrices. These matrices have important applications in various disciplines including image processing, communications, signal processing, encoding, solving Toeplitz matrix problems, preconditioner, and solving least squares problems. They have been put on the firm basis with the work of Davis [6] and Jiang and Zhou [7].

The -circulant matrices play an important role in various applications as well; for details please refer to [8, 9] and the references therein. Ngondiep et al. showed the singular values of -circulants in [10]. There are discussions about the convergence in probability and in distribution of the spectral norm of -circulant matrices in [11].

For , the Jacobsthal number is defined by , where and . If we generalize to and let , , then we obtain Jacobsthal-Lucas number . Let and be the roots of the characteristic equation ; then the Binet formulas of the sequences and have the form [12, 13]

Besides, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices in [6]. It is worth pointing out that the computational complexity of these algorithms is amazing with the order of matrix increasing. However, some authors gave the explicit determinant and inverse of the circulant and skew-circulant matrix involving Fibonacci and Lucas numbers. For example, authors [14] studied the nonsingularity of the skew-circulant type matrices and presented explicit determinants and inverse matrices of these special matrices. Furthermore, four kinds of norms and bounds for the spread of these matrices are given separately. In [15], the nonsingularity of circulant type matrices with the sum and product of Fibonacci and Lucas numbers is discussed. And the exact determinants and inverses of these matrices are given. In [16], Jiang and Hong presented exact determinants of some special circulant matrices involving four kinds of famous numbers. Jaiswal evaluated some determinants of circulant matrix whose elements are the generalized Fibonacci numbers in [17]. Lind presented the determinants of circulant and skew-circulant matrix involving Fibonacci numbers in [18]. Dazheng gave the determinant of the Fibonacci-Lucas quasicyclic matrices in [19]. Shen et al. considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices in [20].

The purpose of this paper is to obtain the better results for the determinants and inverses of circulant type matrices by some properties of Jacobsthal and Jacobsthal-Lucas numbers.

Definition 1 (see [21]). A right circulant matrix (or simply, circulant matrix) is written as

Definition 2 (see [21]). A left circulant matrix is written as

Definition 3 (see [22]). A -circulant matrix is an complex matrix with the following form: where is a nonnegative integer and each of the subscripts is understood to be reduced modulo .

The first row of is ; its th row is obtained by giving its th row a right circular shift by positions (equivalently, mod positions). Note that or yields the standard circulant matrix. If , then we obtain the left circulant matrix.

2. Circulant Matrix with the Jacobsthal Numbers

In this section, let be a circulant matrix. Firstly, we give a determinant formula for the matrix . Afterwards, we prove that is an invertible matrix for , and then we find the inverse of the matrix .

Theorem 4. Let be a circulant matrix; then we have where is the th Jacobsthal number.

Proof. Obviously, when , satisfies the formula (5). In the case , let be two matrices; then the product has the form where So we obtain while and we have The proof is completed.

Theorem 5. Let be a circulant matrix. If , then is an invertible matrix.

Proof. When in Theorem 4, then we have ; hence is invertible. In the case , since , where ,  , we have If there exists    such that , then we obtain for ; thus, is a real number. While , so we have for . But is not the root of the equation   . We obtain for any   , while . By Lemma 4 in [15], the proof is completed.

Lemma 6. Let the matrix be of the form and then the inverse is equal to

Proof. Let . Obviously, for . In the case , we obtain For , we obtain Hence, we verify , where is identity matrix. Similarly, we can verify . Thus, the proof is completed.

Theorem 7. Let    be a circulant matrix; then one has where

Proof. Let where Then we have where is a diagonal matrix, and is the direct sum of and . If we denote , then we obtain since the last row entries of the matrix are .  By Lemma 6, if we let , then its last row entries are given by the following equations: Let , ; then we have We obtain where The proof is completed.

3. Circulant Matrix with the Jacobsthal-Lucas Numbers

In this section, let be a circulant matrix. Firstly, we give a determinant formula for the matrix . Then, we prove that is an invertible matrix for any positive integer , and finally we find the inverse of the matrix .

Theorem 8. Let be a circulant matrix; then one has where is the th Jacobsthal-Lucas number.

Proof. Obviously, when , satisfies the formula (28). When , let be two matrices; then the product has the form where We obtain while and we have

Theorem 9. Let be a circulant matrix; then is invertible for any positive integer .

Proof. Since , where ,  . We have If there exists    such that , we obtain for ; thus, is a real number. While , so we have for . But is not the root of the equation for any positive integer . We obtain for any   , while . By Lemma 4 in [15], the proof is completed.

Lemma 10. Let the matrix be of the form and then the inverse of the matrix is equal to

Proof. Let . Obviously, for . In the case , we obtain For , we obtain Hence, we verify , where is identity matrix. Similarly, we can verify . Thus, the proof is completed.

Theorem 11. Let be a circulant matrix; then we have where

Proof. Let where We have where is a diagonal matrix and is the direct sum of and . If we denote , then we obtain since the last row entries of the matrix are . By Lemma 10, if we let , then its last row entries are given by the following equations: Let and we have We obtain where

4. Left Circulant Matrix with the Jacobsthal and Jacobsthal-Lucas Numbers

In this section, let and be two left circulant matrices. By using the obtained conclusions, we give the determinant for the matrices and . Then, we prove that is an invertible matrix for and is an invertible matrix for any positive integer . The inverses of the matrices and are also presented.

According to Lemma 5 in [15] and Theorems 4, 5, and 7, we can obtain the following theorems.

Theorem 12. Let be a left circulant matrix; then one can prove that

Theorem 13. Let be a left circulant matrix; if , then is an invertible matrix.

Theorem 14. Let , be a left circulant matrix; then one has where

By Lemma 5 in [15] and Theorems 8, 9, and 11, the following conclusions can be obtained.

Theorem 15. Let be a left circulant matrix; then one has where is the th Jacobsthal-Lucas number.

Theorem 16. Let be a left circulant matrix; then is invertible for any positive integer .

Theorem 17. Let be a left circulant matrix; then one has where

5. -Circulant Matrix with the Jacobsthal and Jacobsthal-Lucas Numbers

In this section, let and be two -circulant matrices. By using the obtained conclusions, we give a determinant formula for the matrices and . Then, we prove that is an invertible matrix for and is an invertible matrix for any positive integer if . The inverses of the matrices and are also presented.

From Lemmas 6 and 7 in [15] and Theorems 4, 5, and 7, we deduce the following results.

Theorem 18. Let be a -circulant matrix. Then one has where is the th Jacobsthal number.

Theorem 19. Let be a circulant matrix and , when , is an invertible matrix.

Theorem 20. Let    be a circulant matrix and ; one has where

Taking Lemmas 6 and 7 in [15] and Theorems 8, 9, and 11 into account, we have the following theorems.

Theorem 21. Let be a circulant matrix; then one has where is the th Jacobsthal-Lucas number.

Theorem 22. Let be a -circulant matrix and ; then is invertible for any positive integer .

Theorem 23. Let be a -circulant matrix and ; then one has where

6. Conclusion

This paper obtains better results for the determinants and inverses of circulant type matrices by some properties of Jacobsthal and Jacobsthal-Lucas numbers. The reason we focus our attentions on circulant type matrices is to explore the application of circulant type matrices in solving fractional order differential equations.

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 NSFC (Grant no. 11301252) and the AMEP of Linyi University, China.