Invertibility and Explicit Inverses of Circulant-Type Matrices with 𝑘-Fibonacci and 𝑘-Lucas Numbers

Abstract

Circulant matrices have important applications in solving ordinary differential equations. In this paper, we consider circulant-type matrices with the -Fibonacci and -Lucas numbers. We discuss the invertibility of these circulant matrices and present the explicit determinant and inverse matrix by constructing the transformation matrices, which generalizes the results in Shen et al. (2011).

1. Introduction

Circulant matrices play an important role in solving ordinary differential equations. Wilde [1] 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. By using the well-known results on circulant matrices for computation of eigenvalues of the perturbation coefficient matrix, in computing the stability criteria, Voorhees and Nip [2] got that the rate constants were all set equal to one. By using a Strang-type block-circulant preconditioner, Zhang et al. [3] speeded up the convergent rate of boundary-value methods. Joy and Tavsanoglu [4] showed that feedback matrices of ring cellular neural networks, which can be described by the ODE, are block circulants. Delgado et al. [5] developed some techniques to obtain global hyperbolicity for a certain class of endomorphisms of with ; this kind of endomorphisms is obtained from vectorial difference equations where the mapping defining these equations satisfies a circulant matrix condition. In [6], 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.

Circulant-type matrices include the circulant and left circulant and -circulant matrices. They have been put on the firm basis with the work of Davis [7], Gray [8], and Jiang and Zhou [9]. In [10], the authors pointed out the processes based on the eigenvalue of the 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 [11]. Furthermore, the -circulant matrices are focused on by many researchers; for details, please refer to [12, 13] and the references therein. Ngondiep et al. showed the singular values of -circulants in [14].

The k-Fibonacci and k-Lucas number sequences are defined by the following recurrence relations [15, 16], respectively,

Let and be the roots of the characteristic equation ; then the Binet formulas of the sequences and have the form where .

Besides, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [7, 9]. It is worth pointing out that the computational complexity of these algorithms is very amazing with the order of matrix increasing. However, some authors gave the explicit determinant and inverse of the circulant and skew-circulant involving Fibonacci and Lucas numbers. Shen et al. considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses in [17]. Cambini presented an explicit form of the inverse of a particular circulant matrix in [18]. Bozkurt and Tam gave determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers in [19].

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

In this paper, we adopt the following two conventions and, for any sequence in the case . (1)Circulant matrix: the circulant matrix (denoted by ) with input is the matrix whose th entry is . (2)Left circulant matrix: this is also a symmetric matrix (denoted by ) where the th element of the matrix is . (3)-circulant matrix: a -circulant matrix is an matrix as in 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, positions). Note that or yields the standard circulant matrix. If , then we obtain the so-called left circulant matrix.

Lemma 1 (see [7, 9]). Let be a circulant matrix; then we have the following. (i) is invertible if and only if , , where and ; (ii)If is invertible, then the inverse of is a circulant matrix.

Lemma 2. Let ; the matrix is an orthogonal cyclic shift matrix. It holds that

Lemma 3 (see [20]). The matrix is unitary if and only if , where is a -circulant matrix with the first row .

Lemma 4 (see [20]). is a -circulant matrix with the first row if and only if , where .

Lemma 5 (see [20]). The inverse of a nonsingular -circulant matrix is an -circulant , where satisfies , for some integer .

2. Circulant Matrix with the -Fibonacci Numbers

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

In the following, let , , , and .

Theorem 6. Let be a circulant matrix; then we have where is the th -Fibonacci number. Specially, when , this result is the same as Theorem  2.1 in [17].

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

Theorem 7. Let be a circulant matrix; if , then is an invertible matrix. Specially, when , we get Theorem  2.2 in [17].

Proof. When in Theorem 6, 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 hence, , so we have for . But is not the root of the equation . Hence, we obtain for any , while . Hence, by Lemma 1, the proof is completed.

Lemma 8. Let the matrix be of the form Then the inverse of the matrix is equal to
In particular, when , we get Lemma  2.1 in [17].

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

Theorem 9. Let be a circulant matrix; then we have where . Specially, when , this result is the same as Theorem in [17].

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 elements of matrix are , , by Lemma 8, if , then its last row elements are given by the following equations:
Let ; then we have
Hence, we obtain

3. Circulant Matrix with the -Lucas 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 any positive integer , and then we find the inverse of the matrix .

Theorem 10. Let be a circulant matrix; then we have where is the th -Lucas number. In particular, when , we get Theorem  3.1 in [17].

Proof. Obviously, satisfies formula (26); when , let be two matrices; then the form of is given as follows: where Hence, we obtain while thus, we have

Theorem 11. Let be a circulant matrix; then is invertible for any positive integer . Specially, when , we get Theorem  3.2 in [17].

Proof. Since , where , , we have
If there exists such that , then we obtain for ; thus, is a real number. While hence, , so we have for . But is not the root of the equation for any positive integer . Hence, we obtain for any , while . Thus, by Lemma 1, the proof is completed.

Lemma 12. Let the matrix be of the form then the inverse of the matrix is equal to
Specially, when , we get Lemma  3.1 in [17].

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

Theorem 13. Let be a circulant matrix; then we have
In particular, when , the result is the same as Theorem  3.3 in [17].

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 elements of the matrix are by Lemma 12, if , then its last row elements are given by the following equations: Let ; then we have Hence, we obtain

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

In this section, let and be left 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 for any positive integer . The inverse of the matrices and is also presented.

According to Lemma 2, Theorem 6, Theorem 7, and Theorem 9, we can obtain the following theorems.

Theorem 14. Let be a left circulant matrix; then we have where is the th -Fibonacci number.

Theorem 15. Let be a left circulant matrix; if , then is an invertible matrix.

Theorem 16. Let be a left circulant matrix; then we have where

By Lemma 2, Theorem 10, Theorem 11, and Theorem 13, the following conclusions can be attained.

Theorem 17. Let be a left circulant matrix; then we have where is the th -Lucas number.

Theorem 18. Let be a left circulant matrix; then is invertible for any positive integer .

Theorem 19. Let be a left circulant matrix; then we have

5. -Circulant Matrix with the -Fibonacci and -Lucas 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 is also presented.

From Lemmas 3, 4, and 5 and Theorem 6, Theorem 7, and Theorem 9, we deduce the following results.

Theorem 20. Let - be a -circulant matrix and ; then we have where is the th -Fibonacci number.

Theorem 21. Let - be a -circulant matrix and ; if , then is an invertible matrix.

Theorem 22. Let - be a -circulant matrix and ; then we have where .

Taking Lemmas 3, 4, and 5 and Theorem 10, Theorem 11 and ,Theorem 13 into account, we have the following theorems.

Theorem 23. Let - be a -circulant matrix and ; then we have where is the th -Lucas number.

Theorem 24. Let - be a -circulant matrix and ; then is invertible matrix.

Theorem 25. Let - be a -circulant matrix and ; then we have

6. Conclusion

Circulant-type matrices have a very nice structure, and the -Fibonacci and -Lucas numbers also have amazing properties. The related problem of circulant-type matrices and some famous numbers are studied in this paper. We not only study invertibility of circulant-type matrices with the -Fibonacci and -Lucas numbers but also give the explicit determinants and explicit inverses. We would get a lot of good results if we combine famous numbers with circulant-type matrices, and the results would be used in solving ordinary differential equations. We will still focus our attentions on specific matrices and famous numbers in the future.

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. 11301251) and the AMEP of Linyi University, China.