Circulant Type Matrices with the Sum and Product of Fibonacci and Lucas Numbers

Zhaolin Jiang,Yun Gao,Yanpeng Gong

doi:10.1155/2014/375251

Zhaolin Jiang, Yun Gao + Show 1 more

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https://doi.org/10.1155/2014/375251

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Abstract

Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant andg-circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. 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, andg-circulant matrices and circulant matrix, respectively.

Highlights

Circulant matrices may play a crucial role for solving various differential equations
Afterwards, we prove that An is an invertible matrix for n > 2, and we find the inverse of the matrix An
Afterwards, we prove that Bn is an invertible matrix for any positive integer n, and we find its inverse

Summary

Introduction

Circulant matrices may play a crucial role for solving various differential equations. Matrix and partial differential equations involving factor circulants are considered. Some authors gave the explicit determinants and inverse of circulant and skewcirculant involving Fibonacci and Lucas numbers. Shen et al considered circulant matrices with Fibonacci and Lucas numbers and presented their explicit determinants and inverses by constructing the transformation matrices [21]. The purpose of this paper is to obtain the explicit determinants and inverse of circulant type matrices by some perfect properties of Fn and Ln. In this paper, we adopt the following two conventions 00 = 1, and for any sequence {an}, ∑nk=i ak = 0 in the case i > n. By Lemma 10, if we let A−n1 = Circ(x1, x2, . By Lemma 14, if we let B−n1 = Circ(y1, y2, . (2Lj−i+4 − Lj−i+5) (Ln (L1 − Ln+1)i (2Lj−i+3 − Lj−i+4) (Ln (L1 − Ln+1)i (2Lj−i+2 − Lj−i+3) (Ln (L1 − Ln+1)i (2L2 − L3) (Ln − 2)j+1 (L1 − Ln+1)j+2

Determinant and Inverse of a Left Circulant Matrix with Fn and Ln Numbers

Determinant and Inverse of g-Circulant Matrix with Fn and Ln Numbers