Abstract

It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant, -circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonacci -circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

1. Introduction

Circulant type matrices have important applications in various networks engineering. Exploiting the circulant structure of the channel matrices, Eghbali et al. [1] analysed the realistic near fast fading scenarios with circulant frequency selective channels. The optimum sampling in the one- and two-dimensional (1D and 2D) wireless sensor networks (WSNs) with spatial temporally correlated data was studied with circulant matrices in [2]. The repeat space theory (RST) was extended to apply to carbon nanotubes and related molecular networks, where the corresponding matrices are pseudocirculant in [3]. Preconditioners obtained by circulant approximations of stochastic automata networks were considered in [4]. In [5], circulant mutation whose differential equations obtained neither are of repliator-type nor can they be transformed straightway into a linear equation was introduced into autocatalytic reaction networks. Jing and Jafarkhani [6] proposed distributed differential space-time codes that work for networks with any number of relays using circulant matrices. Wang and Cheng [7] studied the existence of doubly periodic travelling waves in cellular networks involving the discontinuous Heaviside step function by circulant matrix. Pais et al. [8] proved conditions for the existence of stable limit cycles arising from multiple distinct Hopf bifurcations of the dynamics in the case of circulant fitness matrices.

Circulant type matrices have been put on the firm basis with the work in [9, 10] and so on. Furthermore, the -circulant matrices are focused on by many researchers; for the details please refer to [11–13] and the references therein.

Lately, some scholars gave the explicit determinant and inverse of the circulant and skew-circulant involving famous numbers. Jiang et al. [14] discussed the invertibility of circulant type matrices with the sum and product of Fibonacci and Lucas numbers and presented the determinants and the inverses of these matrices. Jiang et al. [15] considered circulant type matrices with the -Fibonacci and -Lucas numbers and presented the explicit determinant and inverse matrix by constructing the transformation matrices. Jiang and Hong [16] gave exact form determinants of the RSFPLR circulant matrices and the RSLPFL circulant matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas number by the inverse factorization of polynomial. Bozkurt and Tam gave determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers in [17]. Cambini presented an explicit form of the inverse of a particular circulant matrix in [18]. Shen et al. considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses in [19].

The Tribonacci sequences are defined by the following recurrence relations [20–22], respectively: where , , , .

The first few values of the sequences are given by the following table:Let , , and be the roots of the characteristic equation and then we have Hence the Binet formulas of the sequences have the form where is the th root of the polynomial for .

In this paper, we consider circulant type matrices, including the circulant and left circulant and -circulant matrices. If we suppose is the th Tribonacci number, then we define a Tribonacci circulant matrix which is an matrix with the following form:

Besides, a Tribonacci left circulant matrix is given bywhere each row is a cyclic shift of the row above to the left.

A Tribonacci -circulant matrix is an 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 , and its th row is obtained by giving its th row a right circular shift by positions (equivalently, mod positions). Note that or yields standard Tribonacci circulant matrix. If , then we obtain Tribonacci left circulant matrix.

Lemma 1. The tridiagonal matrix is given bythen

Proof. ; let , and then let , be the roots of the equation .
We havewhere and .
Hence,

Lemma 2. Letbe an matrix; one has where Specifically, .

Proof. Accoding to the last column determinant expansion and Lemma 1, we obtain

2. Determinant and Inverse of Tibonacci Circulant Matrix

In this section, let be a Tribonacci circulant matrix. Firstly, we give the determinant of the matrix . Afterwards, we discuss the invertibility of the matrix , and we find the inverse of the matrix .

Theorem 3. Let be a Tribonacci circulant matrix; then we have where is the th Tribonacci number, and

Proof. Obviously, satisfies (16). In the case where , letbe two matrices and then we havewhere We obtain whereLet be two matrices, and , and .
Aoccording to Lemma 2, thusWhile we have

Theorem 4. Let be a Tribonacci circulant matrix; if and , then is an invertible matrix.

Proof. When in Theorem 3, then we have . Hence, is not invertible.
In the case where , since , where is the th root of the polynomial , we have If there exists such that , we obtain for . Thus, .
Let , , and ; if , we have is a real number.
While . We have for , but . We obtain for any , while .
If , is an imaginary number.
If or we obtain , such that . If , we have and if is an even number, then . By Lemma 1 in [15], the proof is completed.

Lemma 5. Let be an matrix; then where , is a row vector, and is a column vector.

Proof. Consider