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.

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.

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 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 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.

scholarly journals Isomorphic Operators and Functional Equations for the Skew-Circulant Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhaolin Jiang ◽  
Tingting Xu ◽  
Fuliang Lu
Keyword(s):  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Cyclic Group ◽  
Functional Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Linear Vector ◽  
Skew Circulant ◽  
Linear Vector Space

The skew-circulant matrix has been used in solving ordinary differential equations. We prove that the set of skew-circulants with complex entries has an idempotent basis. On that basis, a skew-cyclic group of automorphisms and functional equations on the skew-circulant algebra is introduced. And different operators on linear vector space that are isomorphic to the algebra ofn×ncomplex skew-circulant matrices are displayed in this paper.

scholarly journals Invertibility and Explicit Inverses of Circulant-Type Matrices withk-Fibonacci andk-Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhaolin Jiang ◽  
Yanpeng Gong ◽  
Yun Gao
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inverse Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices

Circulant matrices have important applications in solving ordinary differential equations. In this paper, we consider circulant-type matrices with thek-Fibonacci andk-Lucas numbers. We discuss the invertibility of these circulant matrices and present the explicit determinant and inverse matrix by constructing the transformation matrices, which generalizes the results in Shen et al. (2011).

scholarly journals Determinants, Norms, and the Spread of Circulant Matrices with Tribonacci and Generalized Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Juan Li ◽  
Zhaolin Jiang ◽  
Fuliang Lu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Partial Differential ◽  
Tribonacci Numbers ◽  
Generalized Lucas Numbers

Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degreen, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

scholarly journals VanderLaan Circulant Type Matrices

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Hongyan Pan ◽  
Zhaolin Jiang
Keyword(s):  
Lower Bound ◽  
Complex Systems ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Circulant Matrices ◽  
Control Methods ◽  
Tridiagonal Matrices ◽  
Transformation Matrices

Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, andg-circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaang-circulant matrix.

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