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 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 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 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.

ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

2014 ◽  
Vol 98 (3) ◽  
pp. 289-310 ◽  
Author(s):  
PETER BUNDSCHUH ◽  
KEIJO VÄÄNÄNEN
Keyword(s):  
Rational Function ◽  
Complex Number ◽  
Functional Equations ◽  
Algebraic Independence ◽  
Lucas Numbers ◽  
Mahler Functions ◽  
Independence Results ◽  
Nonzero Complex Number ◽  
Fibonacci And Lucas Numbers ◽  
Reciprocal Sums

This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

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 Exact Determinants of Some Special Circulant Matrices Involving Four Kinds of Famous Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaoyu Jiang ◽  
Kicheon Hong
Keyword(s):  
Differential Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Number

Circulant matrix family is used for modeling many problems arising in solving various differential equations. The RSFPLR circulant matrices and RSLPFL circulant matrices are two special circulant matrices. The techniques used herein are based on the inverse factorization of polynomial. The exact determinants of these matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas number are given, respectively.

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