Abstract
Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant and -circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. 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, and -circulant matrices and circulant matrix, respectively.
1. Introduction
Circulant matrices may play a crucial role for solving various differential equations. In [1], Ruiz-Claeyssen and dos Santos Leal introduced factor circulant matrices: matrices with the structure of circulants, but with the entries below the diagonal being multiplied by the same factor. The diagonalization of a circulant matrix and the spectral decomposition are conveniently generalized to block matrices with the structure of factor circulants. Matrix and partial differential equations involving factor circulants are considered. Wu and Zou in [2] discussed the existence and approximation of solutions of asymptotic or periodic boundary value problems of mixed functional differential equations. They focused on in [2] with a circulant matrix, whose principal diagonal entries are zeroes. In [3], some Routh-Hurwitz stability conditions are generalized to the fractional order case. The authors considered the 1-system CML . They selected a circulant matrix, which reads a tridiagonal matrix. Ahmed and Elgazzar used coupled map lattices (CML) as an alternative approach to include spatial effects in fractional order systems (FOS). Consider the 1-system CML in [4]. They claimed that the system is stable if all the eigenvalues of the circulant matrix satisfy in [4]. Trench considered nonautonomous systems of linear differential equations in [5] with some constraint on the coefficient matrix . One case is that is a variable block circulant matrix. Kloeden et al. adopted the simplest approximation schemes for in [6] with the Euler method, which reads in [6]. 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 . Guo et al. concerned on generic Dn-Hopf bifurcation to a delayed Hopfield-Cohen-Grossberg model of neural networks in [7], where denoted an interconnection matrix. They especially assumed is a symmetric circulant matrix. Lin and Yang discretized the partial integrodifferential equation (PIDE) in pricing options with the preconditioned conjugate gradient (PCG) method, which constructed the circulant preconditioners. By using FFT, the cost for each linear system is , where is the size of the system in [8]. Lee et al. investigated a high-order compact (HOC) scheme for the general two-dimensional (2D) linear partial differential equation in [9] with a mixed derivative. Meanwhile, in order to establish the 2D combined compact difference (CCD2) scheme, they rewrote in [9] into in [9]. To write the CCD2 system in a concise style, they employed circulant matrix to obtain the corresponding whole CCD2 linear system in [9], whose entries are circulant block.
Circulant type matrices have important applications in various disciplines including image processing, communications, signal processing, encoding, solving Toeplitz matrix problems, and least squares problems. They have been put on firm basis with the work of Davis [10], Jiang and Zhou [11], and Gray [12].
In [13], the authors pointed out the processes based on the eigenvalue of circulant type matrices with i.i.d. entries. There are discussions about the convergence in probability and in distribution of the spectral norm of circulant type matrices in [14]. The -circulant matrices play an important role in various applications as well. For details, please refer to [15, 16] and the references therein. Ngondiep et al. showed the singular values of -circulants in [17]. In [18, 19], the authors gave the limiting spectral distributions of left circulant matrices.
The Fibonacci and Lucas sequences are defined by the following recurrence relations [20, 21], respectively:
For , the first few values of the sequences are given by the following equation:
Let , be the roots of characteristic equation ; then the Binet formulas of the sequences and have the form
Let and , so we can get two new sequences and [22]. The two sequences are defined by the following recurrence relations, respectively:
For , the first few values of the sequences are given by the following equation:
The is given by the formula , where , are the roots of . is given by the formula , where , are the roots of .
Besides, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [10, 11]. Unfortunately, the computational complexities of these algorithms are very amazing with the order of matrix increasing. However, some authors gave the explicit determinants and inverse of circulant and skew-circulant involving Fibonacci and Lucas numbers. For example, Dazheng gave the determinant of the Fibonacci-Lucas quasicyclic matrices in [20]. Shen et al. considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices [21]. Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers [23]. Lind presented the determinants of circulant and skew-circulant involving Fibonacci numbers [24]. Bozkurt and Tam gave determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers [25].
In [22], the authors gave some determinantal and permanental representations of and and complex factorization formulas. The purpose of this paper is to obtain the explicit determinants and inverse of circulant type matrices by some perfect properties of and .
In this paper, we adopt the following two conventions , and for any sequence , in the case .
Definition 1 (see [10, 11]). In a circulant matrix (or right circulant matrix [26]) each row is a cyclic shift of the row above to the right.
Circulant matrix is a special case of a Toeplitz matrix. It is evidently determined by its first row (or column).
Definition 2 (see [11, 26]). In a left circulant matrix (or reverse circulant matrix [13, 14, 18, 19]) each row is a cyclic shift of the row above to the left.
Left circulant matrix is a special Hankel matrix.
Definition 3 (see [14, 27]). 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.
Lemma 4 (see [21]). Let be a circulant matrix; then one has(i)is invertible if and only if , , where and ;(ii)If is invertible, then the inverse of is a circulant matrix.
Lemma 5. Define the matrix is an orthogonal cyclic shift matrix (and a left circulant matrix). It holds that .
Lemma 6 (see [27]). The matrix is unitary if and only if , where is a -circulant matrix with the first row .
Lemma 7 (see [27]). is a -circulant matrix with the first row if and only if , where .
2. Determinant and Inverse of a Circulant Matrix with the Product of the Fibonacci and Lucas Numbers
In this section, let be a circulant matrix. Firstly, we give the determinant equation of matrix . Afterwards, we prove that is an invertible matrix for , and then we find the inverse of the matrix .
Theorem 8. Let be a circulant matrix; then one has where is the th number.
Proof. Obviously, satisfies (10). In the case , let We can obtain where We obtain while we have Thus, the proof is completed.
Theorem 9. Let be a circulant matrix; if , then is an invertible matrix.
Proof. When , in Theorem 8, we have ; hence is invertible. In the case , since , where , .We have
If there exists such that , we obtain for ; thus, is a real number. While , hence, , so we have for . But is not the root of equation . We obtain for any , while . By Lemma 4, 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 We verify , where is the identity matrix. Similarly, we can verify . Thus, the proof is completed.
Theorem 11. Let be a circulant matrix; then one has 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 elements of the matrix are
By Lemma 10, if we let , its last row elements are given by the following equations:
Let ; we have
We obtain
3. Determinant and Inverse of a Circulant Matrix with the Sum of the Fibonacci and Lucas Numbers
In this section, let be a circulant matrix. Firstly, we give an explicit determinant formula of matrix . Afterwards, we prove that is an invertible matrix for any positive integer , and then we find its inverse.
Theorem 12. Let be a circulant matrix; then one has where is the th number.
Proof. Obviously, satisfies (31), when . Let
Then
where
We can obtain
while
We have
Theorem 13. Let be a circulant matrix; then is invertible for any positive integer .
Proof. Since , where , . We have
If there exist such that , we obtain for ; is a real number, while .
Hence, , so we have for . But is not the root of the equation for any positive integer . We obtain for any , while . By Lemma 4, the proof is completed.
Lemma 14. Let matrix be of the form and then inverse of the matrix is equal to
Proof. Let . Obviously, for . In the case , we obtain For , we obtain We verify , where is the identity matrix. Similarly, we can verify . Thus, the proof is completed.
Theorem 15. Let be a circulant matrix; then one has 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 elements of the matrix are
By Lemma 14, if we let , then its last row elements are given by the following equations:
Let ; we have
We obtain
4. Determinant and Inverse of a Left Circulant Matrix with and Numbers
In this section, let and be left circulant matrices. By using the obtained conclusions, we give a determinant formula for the matrix and . Afterwards, 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 and Theorems 8, 9, and 11, we can obtain the following theorems.
Theorem 16. Let be a left circulant matrix; then one has where is the th number.
Theorem 17. Let be a left circulant matrix; if , then is an invertible matrix.
Theorem 18. Let be a left circulant matrix; then one has where
By Lemma 5 and Theorems 12, 13, and 15, the following conclusions can be attained.
Theorem 19. Let be a left circulant matrix; then one has where is the th number.
Theorem 20. Let be a left circulant matrix; then is invertible for any positive integer .
Theorem 21. Let be a left circulant matrix; then one can obtain where
5. Determinant and Inverse of -Circulant Matrix with and Numbers
In this section, let - and - be -circulant matrices. By using the obtained conclusions, we give a determinant formula for the matrices and . Afterwards, we prove that is an invertible matrix for and is an invertible matrix if . The inverse of the matrices and are also presented.
From Lemmas 6 and 7 and Theorems 8, 9, and 11, we deduce the following results.
Theorem 22. Let - be a -circulant matrix; then one has where is the th number.
Theorem 23. Let - be a -circulant matrix and ; if , then is an invertible matrix.
Theorem 24. Let - be a -circulant matrix and ; then where
Taking Lemmas 6 and 7 and Theorems 12, 13, and 15 into account, one has the following theorems.
Theorem 25. Let - be a -circulant matrix; then one has where is the number.
Theorem 26. Let - be a -circulant matrix and ; then is invertible for any positive integer .
Theorem 27. Let - be a -circulant matrix and ; then where
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 & Technology of Shandong Province (Grant no. 2012GGX10115) and NSFC (Grant no. 11301252) and the AMEP of Linyi University, China.