Determinants of the RFMLR Circulant Matrices with Perrin, Padovan, Tribonacci, and the Generalized Lucas Numbers

Abstract

The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences.

1. Introduction

Several special matrices arise frequently in many fields including image processing, communications, medicine, and signal encoding [1]. The application of a block-circulant matrix approach for singular value decomposition rendered the analysis independent of tracer arrival time to improve the results in [2]. Yin et al. introduced fast algorithms for reconstructing signals from incomplete Toeplitz and circulant measurements and showed that Toeplitz and circulant matrices not only were as effective as random matrices for signal encoding but also permitted much faster decoding in [3]. Wu et al. proposed a technique that was made time-shift insensitive by the use of a block-circulant matrix for deconvolution with (oSVD) and without (cSVD) minimization of oscillation of the derived residue function in [4].

The circulant matrices [5, 6], a fruitful subject of research, have in recent years been extended in many directions. The -circulant matrices are another natural extension of this well-studied class and can be found in [7–10]. The -circulant matrix has a wide application, especially on the generalized cyclic codes [7]. The properties and structures of the -circulant matrices, which are called the row first-minus-last right (RFMLR) circulant matrices, are better than those of the general -circulant matrices, so it is significant that we give our attention to them. We first introduce the definitions of the row first-minus-last right (RFMLR) circulant matrices and row last-minus-first left () circulant matrices. As regards their more properties, please refer to [11, 12].

Definition 1. A row first-minus-last right () circulant matrix with the first row , denoted by , is meant to be a square matrix of the form
We define matrix as the basic RFMLR circulant matrix; that is,

Definition 2. A row last-minus-first left (RLMFL) circulant matrix with the first row , denoted by RLMFLcircfr, is meant to be a square matrix of the form
Let and . By explicit computation, we find where is the backward identity matrix of the form
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 [13]. Lin gave the determinant of the Fibonacci-Lucas quasi-cyclic matrices [14]. Lind presented the determinants of circulant and skew circulant involving Fibonacci numbers in [15]. Shen et al. [16] discussed the determinant of circulant matrix involving Fibonacci and Lucas numbers. Akbulak and Bozkurt [17] gave the norms of Toeplitz involving Fibonacci and Lucas numbers.
The determinant problems of the row first-minus-last right (RFMLR) circulant matrices and row last-minus-first left (RLMFL) circulant matrices involving the Perrin, Padovan, Tribonacci, and the generalized Lucas sequences are considered in this paper. The exact formulae of determinants are presented by using some terms of these sequences. The techniques used herein are based on the inverse factorization of polynomial.
The Perrin and Padovan sequences and [18–20] are defined by a third-order recurrence: with the initial conditions , , and , and , , and .
The Tribonacci and the generalized Lucas sequences and [20, 21] are defined by a third-order recurrence: with the initial conditions , , and and , , and .
The first few members of these sequences are given as follows: Recurrences (6) and (7) involve the characteristic equation . If its roots are denoted by , , , then the following equalities can be derived: Moreover, the Binet form for the Perrin sequence is and the Binet form for Padovan sequence is where Recurrences (8) as well imply the characteristic equation . If its roots are denoted by , , , then we have Furthermore, an exact expression for the th Tribonacci number can be given explicitly by This can be written in a slightly more concise form (the Binet form) as where is the th root of the polynomial . And the Binet form for the generalized Lucas sequence is

2. Main Results

By Proposition 5.1 in [7] and properties of RFMLR circulant matrices [12], we deduce the following lemma.

Lemma 3. Let and be the roots of the characteristic equation of . Then the eigenvalues of are given by and the determinant of is given by

Lemma 4. Suppose are the roots of the characteristic equation of  . If , then where , , and If , then where , and , , are the roots of the equation .

Proof. Since are the roots of the characteristic equation of , can be factored as Let , , be the roots of the equation . If , please see [12] for details of the proof. If , then Let . We derive from . Taking the relation of roots and coefficients into account, we deduce that The proof is completed.

We present the exact formulae of determinants of the RFMLR and RLMFL circulant matrices involving four kinds of famous numbers and the detailed process.

2.1. Determinants of the RFMLR and RLMFL Circulant Matrix Involving Perrin Sequence

Theorem 5. Let = . If is odd, then and if is even, then where where , , and , , are the roots of the equation , , respectively.

Proof. Obviously, has the form In the light of Lemma 3 and the Binet form (11) and (10), we have
By Lemma 4 and recurrence (6), we obtain where and , , are the roots of the equation . And where and , , are the roots of the equation . Consequently, if is odd, then and if is even, then where The proof is completed.

Theorem 6. Let = . Then where

Proof. The matrix has the form According to Lemma 3 and the Binet form (11) and (10), we have Using Lemma 4 and recurrence (6), we obtain where Therefore, The proof is completed.

Corollary 7. Let . If or , then and if or , then where , , , are defined the same as Theorem 6.

Proof. Since the result can be derived from Theorem 6 and relation (4).

2.2. Determinants of the RFMLR and RLMFL Circulant Matrix Involving Padovan Sequence

Theorem 8. Let . If is odd, then and if is even, then where where , , are the roots of the equation and is defined as Theorem 5.

Proof. The matrix has the form The determinant of is from Lemma 3 and the Binet form (12) and (10).
Using Lemma 4 and recurrence (7), we obtain where , , , are the roots of the equation . According to (33), we have the following results: if is odd, then and if is even, then where and is defined as Theorem 5.

Theorem 9. Let . If is odd, then and if is even, then where where , , are the roots of the equation .

Proof. The matrix has the form According to Lemma 3 and the Binet form (12) and (10), we have Using Lemma 4 and (12), we obtain where , , , are the roots of the equation . Employing (43), we have the following results: if is odd, then and if is even, then where

Corollary 10. Let = RLMFLcircfr. If , then and if , then and if , then and if , then where , , are defined the same as Theorem 9.

Proof. The theorem can be proved by using Theorem 9 and relation (4).

2.3. Determinants of the RFMLR and RLMFL Circulant Matrix Involving Tribonacci Numbers

Theorem 11. Let = . If is odd, then and if is even, then where where , , and , , are the roots of the equations , , respectively.

Proof. Obviously, has the form According to Lemma 3 and the Binet form (16) and (14), we have
From Lemma 4 it follows that where , , , are the roots of the equation . And where , , , are the roots of the equation . Consequently, we have the following results: if is odd, then and if is even, then where The proof is completed.

Theorem 12. Let . If is odd, then and if is even, then where where , , are the roots of the equation .

Proof. The matrix has the form According to Lemma 3 and the Binet form (16) and (14), we have Considering Lemma 4 and (17), we obtain where , , , are the roots of the equation . And Consequently, if is odd, then and if is even, then where

Corollary 13. Let = RLMFLcircfr. If  , then and if  (mod 4), then and if  (mod 4), then and if  (mod 4), then where are defined as Theorem 12.

Proof. The theorem can be proved by using Theorem 12 and relation (4).

2.4. Determinants of the RFMLR and RLMFL Circulant Matrix Involving Generalized Lucas Numbers

Theorem 14. Let . If is odd, then and if is even, then where where , , are the roots of the equation , and is defined as Theorem 11.

Proof. The matrix has the form According to Lemma 3 and the Binet form (17) and (14), we have From Lemma 4 and (17), we obtain where , , , are the roots of the equation . And where , , , are the roots of the equation . Hence, if is odd, then and if is even, then where and is defined as Theorem 11.

Theorem 15. Let . If is odd, then and if is even, then where where , , are the roots of the equation , and is defined as Theorem 12.

Proof. The matrix has the form According to Lemma 3, (17), and (14), we have By Lemma 4 and the Binet form (17), we obtain where , , , are the roots of the equation . And Thus, if is odd, then and if is even, then where and is defined as Theorem 12.

Corollary 16. Let = RLMFLcircfr. If  (mod 4), then and if  (mod 4), then and if  (mod 4), then and if  (mod 4), then where , , , are defined as Theorem 15.

Proof. The theorem can be proved by using Theorem 15 and relation (4).

3. Conclusions

The row first-minus-last right (RFMLR) circulant matrices and row last-minus-first left (RLMFL) circulant matrices are two kinds of matrices with specific structure. We explored the determinant problem of these two kinds of matrices when their entries are Perrin, Padovan, Tribonacci, and the generalized Lucas sequences, respectively. On the basis of the inverse factorization of polynomial and the third-order recurrence, Binet form, and other properties of these sequences, we present the exact formulae of determinants by some terms of these famous sequences. Chillag has studied some properties and applications of -circulant matrices in [7]. The RFMLR circulant and RLMFL circulant matrices have more explicit structures and better properties than the general -circulant matrices, so they will play more important roles than the general -circulant matrices in some fields of signal encoding, image processing, and so on. That is the reason why we focus our attentions on RFMLR circulant and RLMFL circulant matrices.

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), the NSFC (Grant no. 11301251), and the AMEP of Linyi University, China.