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.

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.

The Travelling Salesman Problem in symmetric circulant matrices with two stripes

2008 ◽  
Vol 18 (1) ◽  
pp. 165-175 ◽  
Author(s):  
IVAN GERACE ◽  
FEDERICO GRECO
Keyword(s):  
Computational Complexity ◽  
Lower Bound ◽  
Lower Bounds ◽  
Travelling Salesman Problem ◽  
Minimum Cost ◽  
Circulant Matrix ◽  
Upper And Lower Bounds ◽  
Circulant Matrices ◽  
Travelling Salesman ◽  
New Construction

The Symmetric Circulant Travelling Salesman Problem asks for the minimum cost tour in a symmetric circulant matrix. The computational complexity of this problem is not known – only upper and lower bounds have been determined. This paper provides a characterisation of the two-stripe case. Instances where the minimum cost of a tour is equal to either the upper or lower bound are recognised. A new construction providing a tour is proposed for the remaining instances, and this leads to a new upper bound that is closer than the previous one.

scholarly journals Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with the Generalized Tribonacci Numbers

2021 ◽  
pp. 131-151
Author(s):  
Yüksel Soykan
Keyword(s):  
Circulant Matrix ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Euclidean Norm ◽  
Hadamard Products ◽  
Tribonacci Numbers

In this paper, we obtain explicit Euclidean norm, eigenvalues, spectral norm and determinant of circulant matrix with the generalized Tribonacci (generalized (r, s, t)) numbers. We also present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix. Moreover, we give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers.

scholarly journals On k-circulant matrices with the generalized third-order Pell numbers

2021 ◽  
Vol 27 (4) ◽  
pp. 187-206
Author(s):  
Yüksel Soykan ◽  
Keyword(s):  
Circulant Matrix ◽  
Numerical Results ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Euclidean Norm ◽  
Third Order ◽  
Pell Numbers

In this paper, we obtain explicit forms of the sum of entries, the maximum column sum matrix norm, the maximum row sum matrix norm, Euclidean norm, eigenvalues and determinant of k-circulant matrix with the generalized third-order Pell numbers. We also study the spectral norm of this k-circulant matrix. Furthermore, some numerical results for demonstrating the validity of the hypotheses of our results are given.

scholarly journals Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Wenai Xu ◽  
Zhaolin Jiang
Keyword(s):  
Kronecker Product ◽  
Hadamard Product ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Circulant Matrices ◽  
Frobenius Norm

Circulant type matrices have played an important role in networks engineering. In this paper, firstly, some bounds for the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices and Lucas row skew first-minus-last right (RSFMLR) circulant matrices are given. Furthermore, the spectral norm of Hadamard product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is obtained. Finally, the Frobenius norm of Kronecker product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is presented.

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 On Skew Circulant Type Matrices Involving Any Continuous Fibonacci Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zhaolin Jiang ◽  
Jinjiang Yao ◽  
Fuliang Lu
Keyword(s):  
Fibonacci Numbers ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Frobenius Norm ◽  
Transformation Matrices ◽  
Skew Circulant

Circulant and skew circulant matrices have become an important tool in networks engineering. In this paper, we consider skew circulant type matrices with any continuous Fibonacci numbers. We discuss the invertibility of the skew circulant type matrices and present explicit determinants and inverse matrices of them by constructing the transformation matrices. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm and bounds for the spread of these matrices are given, respectively.

scholarly journals On k-circulant matrices with the Lucas numbers

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 4037-4046
Author(s):  
Biljana Radicic
Keyword(s):  
Complex Number ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Circulant Matrices ◽  
Euclidean Norm ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Nonzero Complex Number

Let k be a nonzero complex number. In this paper, we determine the eigenvalues of a k-circulant matrix whose first row is (L1,L2,..., Ln), where Ln is the nth Lucas number, and improve the result which can be obtained from the result of Theorem 7. [28]. The Euclidean norm of such matrix is obtained. Bounds for the spectral norm of a k-circulant matrix whose first row is (L-11, L-12,..., L-1n ) are also investigated. The obtained results are illustrated by examples.

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 Explicit Form of the Inverse Matrices of Tribonacci Circulant Type Matrices

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Li Liu ◽  
Zhaolin Jiang
Keyword(s):  
Explicit Form ◽  
Circulant Matrix ◽  
Inverse Matrix ◽  
Circulant Matrices ◽  
Transformation Matrices

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,g-circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonaccig-circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

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