Skew Circulant Type Matrices Involving the Sum of Fibonacci and Lucas Numbers

Abstract

Skew circulant and circulant matrices have been an ideal research area and hot issue for solving various differential equations. In this paper, the skew circulant type matrices with the sum of Fibonacci and Lucas numbers are discussed. The invertibility of the skew circulant type matrices is considered. The determinant and the inverse matrices are presented. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, the maximum row sum matrix norm, and bounds for the spread of these matrices are given, respectively.

1. Introduction

As is well-known, skew circulant and circulant matrices play a crucial role for solving various differential equations. Authors in [1] presented the skew circulant matrices as preconditioners for linear multistep formulae (LMF-)based ordinary differential equations (ODEs) codes. Claeyssen et al. [2] discussed factor block circulant and periodic solutions of undamped matrix differential equations. Using circulant matrix, Karasözen and Şimşek [3] considered periodic boundary conditions such that no additional boundary terms will appear after semidiscretization. Meyer and Rjasanow [4] have presented an effective direct solution method for certain boundary element equations in 3D. Guo et al. concerned on generic Dn-Hopf bifurcation to a delayed Hopfield-Cohen-Grossberg model of neural networks (5.17) in [5], where denoted an interconnection matrix. In particular, they assumed that is a symmetric circulant matrix. In [6], Jin et al. proposed the GMRES method with the Strang-type block-circulant preconditioner for solving singular perturbation delay differential equations. In [7], two new normal-form realizations are presented which utilize circulant and skew circulant matrices as their state transition matrices. The well known second-order coupled form is a special case of the skew circulant form. Compared with cyclic convolution algorithm, the skew cyclic convolution algorithm [8] is able to perform filtering procedure in approximately half of computational cost for real signals. In [9], a new fast algorithm for optimal design of block digital filters (BDFs) was proposed based on skew circulant matrix. Spectral decompositions of skew circulant and skew left circulant matrices were discussed in [10]. Li et al. [11] gave the style spectral decomposition of skew circulant matrix firstly and then dealt with the optimal backward perturbation analysis for the linear system with skew circulant coefficient matrix.

Some scholars have given various algorithms for the determinants and inverses of nonsingular circulant matrices [12, 13]. Unfortunately, the computational complexity of these algorithms is very amazing with the order of matrix increasing. However, some authors gave the explicit determinants and inverse of circulant and skew circulant involving some famous numbers. For example, Shen et al. considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices in [14]. Gao et al. [15] gave explicit determinants and inverses of skew circulant and skew left circulant matrices with Fibonacci and Lucas numbers. In [16], Jiang et al. discussed 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 [17], Jiang and Hong give exact determinants of some special circulant matrices involving four kinds of famous numbers. Authors [18] discussed the nonsingularity of the circulant type matrix and presented the explicit determinant and inverse matrices.

There are several papers on the norms of some special matrices. Solak [19] established the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries. İpek [20] investigated an improved estimation for spectral norms of these matrices.

Beginning with Mirsky [21] several authors [22–24] have obtained bounds for the spread of a matrix.

Additionally, skew circulant type matrices include skew circulant and skew left circulant matrices. The norm and spread of skew circulant type matrices have not been studied. It is hoped that this paper will help in changing this. More work continuing the present paper is forthcoming.

The sum of Fibonacci and Lucas sequence is defined by the following recurrence relations: for . The first few values of the sequence are given by the following table: The is given by the formula where and are the roots of the characteristic equation .

In this paper, we consider skew circulant type matrices, including the skew circulant and skew left circulant matrices.

We define a sum of Fibonacci and Lucas skew circulant matrix which is an complex matrix with following form: where each row is a cyclic shift of the row above the right.

Besides, a sum of Fibonacci and Lucas skew left circulant matrix is given by where each row is a cyclic shift of the row above the left.

Lemma 1. Let be the sum of Fibonacci and lucas numbers; then,

Proof. According to we have and hence This completes the proof.

Lemma 2 (see [10]). Let be a skew left circulant matrix and let be odd; then, where are the eigenvalues of .

2. Determinant and Inverse of Skew Circulant Matrix with the Sum of Fibonacci and Lucas Numbers

In this section, let be a skew circulant matrix. First of all, a determinant explicit formula for the matrix is given. After that, we prove that is an invertible matrix for any positive interger , and then we find the inverse of the matrix . In the following, let

Theorem 3. Let be a skew circulant matrix; then where is the th sum of Fibonacci and Lucas numbers.

Proof. Obviously, satisfies the equation. In the case , let be two matrices; then, we have where So it holds that While taking , we have This completes the proof.

Theorem 4. Let be a skew circulant matrix; then is an invertible matrix for any positive interger .

Proof. Taking in Theorem 3, we have . Hence is invertible. In the case , since , where , we have where , . If there exists such that , we have for , and hence it follows that is a real number. Since it yields that , so we have for . Since is not the root of the equation . We obtain for any , while It follows from Lemma 1 in [25] that the conclusion holds.

Lemma 5. Let the matrix be of the form Then the inverse of the matrix is equal to

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

Theorem 6. Let be a skew circulant matrix; then where

Proof. Let where Then we have so , and here 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 (), the last row elements of the matrix are (), where Hence it follows from Lemma 5 that by letting then its last row elements are which are given by the following equations: Hence, we obtain where This completes the proof.

3. Norm and Spread of Skew Circulant Matrix with the Sum of Fibonacci and Lucas Numbers

Theorem 7. Let be a skew circulant matrix. The three kinds norms of are given by

Proof. By Definition 4 in [26], (6), and (7), we have Thus

Theorem 8. Let be an odd-order alternative skew circulant matrix and let be odd. Then

Proof. By Lemma 1 in [25], we have Therefore Since is odd, is an eigenvalue of , which is To sum up, we have Since all skew circulant matrices are normal, by Lemma 7 in [26], (6), and (43), we have which completes the proof.

Theorem 9. Let be a skew circulant matrix; then the bounds for the spread of are

Proof. The trace of is denoted by . By Definition 5 in [26] and (36), we have Since by (6) and (8), By Lemma 6 in [26], we have

4. Determinant and Inverse of Skew Left Circulant Matrix with the Sum of Fibonacci and Lucas Numbers

In this section, let be a skew left circulant matrix. By using the obtained conclusions in Section 2, we give a determinant explicit formula for the matrix . And then, we prove that is an invertible matrix for any positive interger . The inverse of the matrix is also presented.

According to Lemmas 4 and 5 in [15] and Theorems 3, 4, and 6, we can obtain the following theorems.

Theorem 10. Let be a skew left circulant matrix; then where is the th sum of Fibonacci and Lucas number.

Theorem 11. Let be a skew left circulant matrix for any positive interger ; then is an invertible matrix.

Theorem 12. Let be a skew left circulant matrix; then where

5. Norm and Spread of Skew Left Circulant Matrix with the Sum of Fibonacci and Lucas Numbers

Theorem 13. Let be a skew left circulant matrix. Then three kinds norms of are given by

Proof. Using the similar method in Theorem 7, the conclusion is obtained.

Theorem 14. Let be an odd-order alternative skew left circulant matrix; then

Proof. According to Lemma 2, for , and So By (56) and (57), we have Since all skew left circulant matrices are symmetrical, by Lemma 7 in [26], (6), and (58), we obtain

Theorem 15. Let be a skew left circulant matrix; then the bounds for the spread of are where .

Proof. Since is a symmetric matrix, by Lemma 6 in [26], we get .
The trace of is, if is odd, then , by (6), we have Let ; then by Definition 5 in [26], (53), and (61), we obtain If is even, then hence So the result follows.

6. Conclusion

We discuss the invertibility of the skew circulant type matrices with the sum of Fibonacci and Lucas numbers and present the determinant and the inverse matrices by constructing the transformation matrices. The four kinds of norms and bounds for the spread of these matrices are given, respectively.

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 the AMEP of Linyi University, China.