#### Abstract

Circulant matrices play an important role in solving delay differential equations. In this paper, circulant type matrices including the circulant and left circulant and -circulant matrices with any continuous Fibonacci and Lucas numbers are considered. Firstly, the invertibility of the circulant matrix is discussed and the explicit determinant and the inverse matrices by constructing the transformation matrices are presented. Furthermore, the invertibility of the left circulant and -circulant matrices is also studied. We obtain the explicit determinants and the inverse matrices of the left circulant and -circulant matrices by utilizing the relationship between left circulant, -circulant matrices and circulant matrix, respectively.

#### 1. Introduction

Circulant matrices have important applications in solving various differential equations [1–3]. The use of circulant preconditioners for solving structured linear systems has been studied extensively since 1986; see [4, 5]. Circulant matrices also play an important role in solving delay differential equations. In [6], Chan et al. proposed a preconditioner called the Strang-type block-circulant preconditioner for solving linear systems from IVPs. The Strang-type preconditioner was also used to solve linear systems from differential-algebraic equations and delay differential equations; see [7–14]. In [15], Jin et al. proposed the GMRES method with the Strang-type block-circulant preconditioner for solving singular perturbation delay differential equations.

The -circulant matrices play an important role in various applications as well; please refer to [16, 17] for details. There are discussions about the convergence in probability and in distribution of the spectral norm of -circulant matrices in [18, 19]. Ngondiep et al. showed the singular values of -circulants in [20].

Recently, some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [21, 22]. Unfortunately, the computational complexity of these algorithms is increasing dramatically with the increasing order of matrices. However, some authors gave the explicit determinants and inverse of circulant involving Fibonacci and Lucas numbers. For example, Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers [23]. Lind presented the determinants of circulant involving Fibonacci numbers [24]. Lin gave the determinant of the Fibonacci-Lucas quasicyclic matrices in [25]. Shen et al. considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses [26]. Bozkurt and Tam gave determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers in [27].

The purpose of this paper is to obtain the explicit determinants, explicit inverses of circulant, left circulant, and -circulant matrices involving any continuous Fibonacci numbers and Lucas numbers. And we generalize the result in [26].

In the following, let be a nonnegative integer. We adopt the following two conventions , and for any sequence in the case .

The Fibonacci and Lucas sequences are defined by the following recurrence relations [23–26], respectively: for . The first few values of the sequences are given by the following table: Let and be the roots of the characteristic equation ; then the Binet formulas of the sequences and have the form

*Definition 1 (see [21, 22]). *In a right circulant matrix (or simply, circulant matrix)
each row is a cyclic shift of the row above to the right. Right circulant matrix is a special case of a Toeplitz matrix. It is evidently determined by its first row (or column).

*Definition 2 (see [22, 28]). *In a left circulant matrix (or reverse circulant matrix )
each row is a cyclic shift of the row above to the left. Left circulant matrix is a special Hankel matrix.

*Definition 3 (see [19, 29]). *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 so called left circulant matrix.

Lemma 4 (see [26]). *Let be circulant matrix; then one has*(i)* is invertible if and only if the eigenvalues of **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 [29]). *The matrix is unitary if and only if , where is a -circulant matrix with first row .*

Lemma 7 (see [29]). * is a -circulant matrix with first row if and only if , where .*

#### 2. Determinant, Invertibility, and Inverse of Circulant Matrix with Any Continuous 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 . Obviously, when , or , is also an invertible matrix.

Theorem 8. *Let be a circulant matrix. Then one has
**
where is the th Fibonacci number. Specially, when , this result is the same as Theorem 2.1 in [26].*

*Proof. *Obviously, satisfies the formula. In the case , let
be two matrices; then we have
where
We obtain
while
we have

Theorem 9. *Let be a circulant matrix; if , then is an invertible matrix. Specially, when , one gets Theorem 2.2 in [26].*

*Proof. *When = 3 in Theorem 8, then 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 the equation . We obtain for any , while . By Lemma 4, the proof is completed.

Lemma 10. *Let the entries of the matrix be of the form
**
then the entries of the inverse of the matrix are equal to
**
In particular, when , one gets Lemma 2.1 in [26].*

*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 one has
**
where
**
Specially, when , this result is the same as Theorem 2.3 in [26].*

*Proof. *Let
where ,
We have
where is a diagonal matrix and is the direct sum of and . If we denote , then we obtain

and the last row elements of the matrix are . By Lemma 10, if let , then its last row elements are given by the following equations:

Let
we have
We obtain
where

#### 3. Determinant, Invertibility, and Inverse of Circulant Matrix with Any Continuous 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 12. *Let be a circulant matrix; then one has
**
where is the th Lucas number. In particular, when , one gets Theorem 3.1 in [26].*

*Proof. *Obviously, satisfies the formula, when ; let
be two matrices, we have
where
We obtain
while
We have

Theorem 13. *Let be a circulant matrix; then is invertible for any positive integer . Specially, when , one gets Theorem 3.2 in [26].*

*Proof. *Since , where , . Hence we have
If there exists such that , we obtain for ; thus, is a real number, while

Hence, ; 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 the entries of the matrix be of the form
**
then the entries of the inverse of the matrix are equal to
**
Specially, when , one gets Lemma 3.1 in [26].*

*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 15. *Let be a circulant matrix; then we have
**
where
**
In particular, when , the result is the same as Theorem 3.3 in [26].*

*Proof. *Let be the form of
where
We have
where is a diagonal matrix and is the direct sum of and . If we denote , we obtain

and the last row elements of the matrix are . By Lemma 14, if let , then its last row elements are given by the following equations:

Let
we have
We obtain