Circulant Type Matrices with the Sum and Product of Fibonacci and Lucas Numbers

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 𝑔 -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 𝑔 -circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and 𝑔 -circulant matrices by utilizing the relation between left circulant, and 𝑔 -circulant matrices and circulant matrix, respectively.


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 (5.13) 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 (10).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 (10) in [4].They claimed that the system is stable if all the eigenvalues of the circulant matrix satisfy (2) in [4].Trench considered nonautonomous systems of linear differential equations (1) 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 (1) in [6] with the Euler method, which reads (5) 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 ( log ) = ( log ).Guo et al. concerned on generic Dn-Hopf bifurcation to a delayed Hopfield-Cohen-Grossberg model of neural networks (5.17) 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 ( log ), where  is the size of the system in [8]. Lee et al. investigated a high-order compact (HOC) scheme for the general twodimensional (2D) linear partial differential equation (1.1) in [9] with a mixed derivative.Meanwhile, in order to establish the 2D combined compact difference (CCD2) scheme, they rewrote (1.1) in [9] into (2.1) in [9].To write the CCD2 system in a concise style, they employed circulant matrix to obtain For  ≥ 0, the first few values of the sequences are given by the following equation: Let F  =   ⋅  and L  =   +  , so we can get two new sequences F  and L  [22].The two sequences are defined by the following recurrence relations, respectively: For  ≥ 0, the first few values of the sequences are given by the following equation: The F  is given by the formula F  = (  1 −   1 )/( 1 −  1 ), where  1 ,  1 are the roots of  2 − 3 + 1 = 0. L  is given by the formula L  =   +   = (  −   )/( − ) + (  +   ), where ,  are the roots of  2 −  − 1 = 0.
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 skewcirculant 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 F  and L  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 F  and L  .
In this paper, we adopt the following two conventions 0 0 = 1, and for any sequence {  }, ∑  =   = 0 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).
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 ( 1 ,  2 , . . .,   ); its ( + 1)th row is obtained by giving its th row a right circular shift by  positions (equivalently,  mod  positions).Note that  = 1 or  =  + 1 yields the standard circulant matrix.If  =  − 1, then we obtain the left circulant matrix.

Determinant and Inverse of a Circulant Matrix with the Product of the Fibonacci and Lucas Numbers
In this section, let A  = Circ(F 1 , F 2 , . . ., F  ) be a circulant matrix.Firstly, we give the determinant equation of matrix A  .Afterwards, we prove that A  is an invertible matrix for  > 2, and then we find the inverse of the matrix A  .
Theorem 8. Let A  = Circ(F 1 , F 2 , . . ., F  ) be a circulant matrix; then one has where F  is the th   ⋅   number.
Proof.Obviously, det A 1 = 1 satisfies (10).In the case  > 1, let We can obtain where We obtain while we have Thus, the proof is completed.

Determinant and Inverse of a Circulant Matrix with the Sum of the Fibonacci and Lucas Numbers
In this section, let B  = Circ(L 1 , L 2 , . . ., L  ) be a circulant matrix.Firstly, we give an explicit determinant formula of matrix B  .Afterwards, we prove that B  is an invertible matrix for any positive integer , and then we find its inverse.
Theorem 12. Let B  = Circ(L 1 , L 2 , . . ., L  ) be a circulant matrix; then one has where L  is the th   +   number.
,  be the roots of characteristic equation  2 −−1 = 0; then the Binet formulas of the sequences {  } and {  } have the form