Abstract and Applied Analysis

Volume 2014 (2014), Article ID 647030, 9 pages

http://dx.doi.org/10.1155/2014/647030

Research Article

Explicit Determinants of the RFPLR Circulant and RLPFL Circulant Matrices Involving Some Famous Numbers

Tingting Xu,1,2 Zhaolin Jiang,¹ and Ziwu Jiang¹

¹Department of Mathematics, Linyi University, Linyi, Shandong 276005, China

²School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China

Received 28 April 2014; Accepted 5 June 2014; Published 19 June 2014

Academic Editor: Tongxing Li

Copyright © 2014 Tingting Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Circulant matrices may play a crucial role in solving various differential equations. In this paper, the techniques used herein are based on the inverse factorization of polynomial. We give the explicit determinants of the RFPLR circulant matrices and RLPFL circulant matrices involving Fibonacci, Lucas, Pell, and Pell-Lucas number, respectively.

1. Introduction

It has been found out that circulant matrices play an important role in solving differential equations in various fields such as Lin and Yang discretized the partial integrodifferential equation (PIDE) in pricing options with the preconditioned conjugate gradient (PCG) method, where constructed the circulant preconditioners. By using the FFT, the cost for each linear system is where is the size of the system in [1]. Lei and Sun [2] proposed the preconditioned CGNR (PCGNR) method with a circulant preconditioner to solve such Toeplitz-like systems. Kloeden et al. adopted the simplest approximation schemes for (1) in [3] with the Euler method, which reads (5) in [3]. 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 . By using a Strang-type block-circulant preconditioner, Zhang et al. [4] speeded up the convergent rate of boundary-value methods. In [5], the resulting dense linear system exhibits so much structure that it can be solved very efficiently by a circulant preconditioned conjugate gradient method. Ahmed et al. used coupled map lattices (CML) as an alternative approach to include spatial effects in FOS. Consider the 1-system CML (10) in [6]. They claimed that the system is stable if all the eigenvalues of the circulant matrix satisfy (2) in [6]. Wu and Zou in [7] 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 [7] with a circulant matrix, whose principal diagonal entries are zeroes.

Circulant matrix family have important applications in various disciplines including image processing, communications, signal processing, encoding, and preconditioner. They have been put on firm basis with the work of Davis [8] and Jiang and Zhou [9]. The circulant matrices, long a fruitful subject of research, have in recent years been extended in many directions [10–13]. The -circulant matrices are another natural extension of this well-studied class and can be found in [14–20]. The -circulant matrix has a wide application, especially on the generalized cyclic codes in [14]. The properties and structures of the -circulant matrices, which are called RFPLR circulant matrices, are better than those of the general -circulant matrices, so there are good algorithms for determinants.

There are many interests in properties and generalization of some special matrices with famous numbers. Jaiswal evaluated some determinants of circulant whose elements are the generalized Fibonacci numbers [21]. Dazheng gave the determinant of the Fibonacci-Lucas quasicyclic matrices [22]. Lind presented the determinants of circulant and skew circulant involving Fibonacci numbers in [23]. Shen et al. [24] discussed the determinant of circulant matrix involving Fibonacci and Lucas numbers. Akbulak and Bozkurt [25] gave the norms of Toeplitz involving Fibonacci and Lucas numbers. The authors [26, 27] discussed some properties of Fibonacci and Lucas matrices. Stanimirović et al. gave generalized Fibonacci and Lucas matrix in [28]. Z. Zhang and Y. Zhang [29] investigated the Lucas matrix and some combinatorial identities.

Firstly, we introduce the definitions of the RFPLR circulant matrices and RLPFL circulant matrices and properties of the related famous numbers. Then, we present the main results and the detailed process.

2. Definition and Lemma

Definition 1. A row first-plus-last -right (RFPLR) circulant matrix with the first row , denoted by , means a square matrix of the form

Note that the RFPLR circulant matrix is a circulant matrix, which is neither an extension nor special case of the circulant matrix [8]. They are two completely different kinds of special matrices.

We define as the basic RFPLR circulant matrix; that is,

Both the minimal polynomial and the characteristic polynomial of are , which has only simple roots, denoted by . In addition, satisfies and . Then a matrix can be written in the form

if and only if is a RFPLR circulant matrix, where the polynomial is called the representer of the RFPLR circulant matrix .

Since is nonderogatory, then is a RFMLR circulant matrix if and only if commutes with ; that is, . Because of the representation, RFMLR circulant matrices have very nice structure and the algebraic properties also can be easily attained. Moreover, the product of two RFMLR circulant matrices and the inverse are again RFMLR circulant matrices.

Definition 2. A row last-plus-first -left (RLPFL) circulant matrix with the first row , denoted by , means a square matrix of the form

Let and . By explicit computation, we find

where is the backward identity matrix of the form

The Fibonacci, Lucas, Pell, and the Pell-Lucas sequences [30–36] are defined by the following recurrence relations, respectively:

The first few values of these sequences are given by the following table ():

The sequences , , , and are given by the Binet formulae

where , are the roots of the characteristic equation and , are the roots of the characteristic equation .

By Proposition 5.1 in [14], we deduce the following lemma.

Lemma 3. Let ; then the eigenvalues of are

and in addition,

where are the roots of the equation

Lemma 4. Consider

where

and satisfy (12), , .

Proof. Consider

while

Since satisfy (12), we must have

So

3. Determinant of the RFPLR and RLPFL Circulant Matrices with the Fibonacci Numbers

Theorem 5. Let . Then

where

Proof. The matrix can be written as

Using Lemma 3, the determinant of is

Using Lemma 4, we obtain

where

Using the method in Theorem 5 similarly, we also have the following.

Theorem 6. Let . Then

Theorem 7. Let . Then

Proof. The matrix can be written as

Hence, we have

where and its determinant is obtained from Theorem 6,

In addition,

so

4. Determinant of the RFMLR and RLMFL Circulant Matrices with the Lucas Numbers

Theorem 8. Let . Then

where

Proof. The matrix can be written as

Using Lemma 3, we have

According to Lemma 4, we obtain

Then, we get

where

Using the method in Theorem 8 similarly, we also have the following.

Theorem 9. Let . Then

where

Theorem 10. Let . Then

where

Proof. The matrix can be written as

Thus, we have

where matrix and its determinant can be obtained from Theorem 9,

where

In addition,

so the determinant of matrix is

where

5. Determinants of the RFPLR and RLPFL Circulant Matrix with the Pell Numbers

Theorem 11. If , then

where

Proof. The matrix can be written as

Using Lemma 3, the determinant of is

According to Lemma 4, we can get

where

Using the method in Theorem 11 similarly, we also have the following.

Theorem 12. If , then

Theorem 13. If , then one has

Proof. The matrix can be written as

Then we can get

where and its determinant could be obtained through Theorem 12; namely,

So

6. Determinants of the RFPLR and RLPFL Circulant Matrix with the Pell-Lucas Numbers

Theorem 14. If , then one has

where

Proof. The method is similar to Theorem 11.

Certainly, we can get the following theorem.

Theorem 15. If , then one gets

where

Theorem 16. If , then

where

7. Conclusion

The determinant problems of the RFPLR circulant matrices and RLPFL circulant matrices involving the Fibonacci, Lucas, Pell, and Pell-Lucas number are considered in this paper. The explicit determinants are presented by using some terms of these numbers.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is 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.

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