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Journal of Applied Mathematics

Volume 2014 (2014), Article ID 585438, 11 pages

http://dx.doi.org/10.1155/2014/585438

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

^{1}Department of Mathematics, Linyi University, Linyi, Shandong 276000, China^{2}Department of Mathematics, Shandong Normal University, Ji’nan, Shandong 250000, China

Received 22 July 2013; Revised 4 December 2013; Accepted 6 December 2013; Published 29 January 2014

Academic Editor: George Jaiani

Copyright © 2014 Zhaolin Jiang 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

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