On Jacobsthal and Jacobsthal-Lucas Circulant Type Matrices

Yanpeng Gong,Zhaolin Jiang,Yun Gao

doi:10.1155/2015/418293

Abstract

Circulant type matrices have become an important tool in solving fractional order differential equations. In this paper, we consider the circulant and left circulant andg-circulant matrices with the Jacobsthal and Jacobsthal-Lucas numbers. First, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix. Furthermore, the invertibility of the left circulant andg-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant andg-circulant matrices by utilizing the relation between left circulant,g-circulant matrices, and circulant matrix, respectively.

Highlights

Circulant type matrices possess interesting properties, which have been exploited to obtain the transient solution in a closed form for fractional order differential equations
The purpose of this paper is to obtain the better results for the determinants and inverses of circulant type matrices by some properties of Jacobsthal and Jacobsthal-Lucas numbers
Afterwards, we prove that An is an invertible matrix for n > 2, and we find the inverse of the matrix An

Summary

Introduction

Circulant type matrices possess interesting properties, which have been exploited to obtain the transient solution in a closed form for fractional order differential equations. Some authors gave the explicit determinant and inverse of the circulant and skewcirculant matrix involving Fibonacci and Lucas numbers. In [15], the nonsingularity of circulant type matrices with the sum and product of Fibonacci and Lucas numbers is discussed. Lind presented the determinants of circulant and skew-circulant matrix involving Fibonacci numbers in [18]. Shen et al considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices in [20]. The purpose of this paper is to obtain the better results for the determinants and inverses of circulant type matrices by some properties of Jacobsthal and Jacobsthal-Lucas numbers.

Circulant Matrix with the Jacobsthal Numbers

Circulant Matrix with the Jacobsthal-Lucas Numbers

Left Circulant Matrix with the Jacobsthal and Jacobsthal-Lucas Numbers

Conclusion