scholarly journals A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jiangming Ma ◽  
Tao Qiu ◽  
Chengyuan He
Keyword(s):  
Fibonacci Numbers ◽  
Matrix Decomposition ◽  
Circulant Matrix ◽  
New Method ◽  
Circulant Matrices ◽  
Computational Time ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

We use a new method of matrix decomposition for r -circulant matrix to get the determinants of A n = Circ r F 1 , F 2 , … , F n and B n = Circ r L 1 , L 2 , … , L n , where F n is the Fibonacci numbers and L n is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Zhaolin Jiang ◽  
Yanpeng Gong ◽  
Yun Gao
Keyword(s):  
Differential Equations ◽  
Circulant Matrix ◽  
Inverse Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Fibonacci And Lucas Numbers

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.

scholarly journals The Invertibility, Explicit Determinants, and Inverses of Circulant and Left Circulant andg-Circulant Matrices Involving Any Continuous Fibonacci and Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Zhaolin Jiang ◽  
Dan Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Fibonacci And Lucas Numbers ◽  
Delay Differential ◽  
The Relationship

Circulant matrices play an important role in solving delay differential equations. In this paper, circulant type matrices including the circulant and left circulant andg-circulant matrices with any continuous Fibonacci and Lucas numbers are considered. Firstly, the invertibility of the circulant matrix is discussed and the explicit determinant and the inverse matrices by constructing the transformation matrices are presented. Furthermore, the invertibility of the left circulant andg-circulant matrices is also studied. We obtain the explicit determinants and the inverse matrices of the left circulant andg-circulant matrices by utilizing the relationship between left circulant,g-circulant matrices and circulant matrix, respectively.

On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers

2020 ◽  
Vol 70 (3) ◽  
pp. 641-656
Author(s):  
Amira Khelifa ◽  
Yacine Halim ◽  
Abderrahmane Bouchair ◽  
Massaoud Berkal
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Difference Equations ◽  
Fibonacci Numbers ◽  
Higher Order ◽  
Real Numbers ◽  
Lucas Numbers ◽  
Rational Difference Equations ◽  
Initial Values ◽  
Fibonacci And Lucas Numbers

AbstractIn this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$where n, k∈ ℕ0, the initial values x−k, x−k+1, …, x0, y−k, y−k+1, …, y0, z−k, z−k+1, …, z1 and z0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

scholarly journals The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jin-jiang Yao ◽  
Zhao-lin Jiang
Keyword(s):  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Skew Circulant ◽  
The Relationship

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

Trigonometry and Fibonacci numbers

2007 ◽  
Vol 91 (521) ◽  
pp. 216-226 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Fibonacci Numbers ◽  
Trigonometric Functions ◽  
Integer Sequences ◽  
Lucas Numbers ◽  
Mathematical Objects ◽  
Lucas Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Trigonometric Identities ◽  
The Way

This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

scholarly journals On the circulant matrices with Ducci sequences and Fibonacci numbers

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5501-5508
Author(s):  
Süleyman Solak ◽  
Mustafa Bahşi ◽  
Osman Kan
Keyword(s):  
Fibonacci Numbers ◽  
Circulant Matrix ◽  
Circulant Matrices

A Ducci sequence generated by A = (a1,a2,...,an)? Zn is the sequence {A,DA,D2A,...} where the Ducci map D : Zn ? Zn is defined by D(A) = D(a1, a2,...,an) = (|a2-a1|, |a3-a2|,..., |an-an-1|, |an-a1|). In this study, we examine some properties of the matrices Cn, DCn, D2Cn; where Cn =Circ(c0,c1,..., cn-1) is a circulant matrix whose entries consist of Fibonacci numbers.

scholarly journals Skew Circulant Type Matrices Involving the Sum of Fibonacci and Lucas Numbers

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Zhaolin Jiang ◽  
Yunlan Wei
Keyword(s):  
Differential Equations ◽  
Research Area ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Frobenius Norm ◽  
Lucas Numbers ◽  
Skew Circulant ◽  
Fibonacci And Lucas Numbers

Skew circulant and circulant matrices have been an ideal research area and hot issue for solving various differential equations. In this paper, the skew circulant type matrices with the sum of Fibonacci and Lucas numbers are discussed. The invertibility of the skew circulant type matrices is considered. The determinant and the inverse matrices are presented. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, the maximum row sum matrix norm, and bounds for the spread of these matrices are given, respectively.

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