3x3 Dimensional Special Matrices Associated with Fibonacci and Lucas Numbers (original) (raw)

A generalization of Fibonacci and Lucas matrices

Discrete Applied Mathematics, 2008

We define the matrix U (a,b,s) n of type s, whose elements are defined by the general second-order non-degenerated sequence and introduce the notion of the generalized Fibonacci matrix F (a,b,s) n , whose nonzero elements are generalized Fibonacci numbers. We observe two regular cases of these matrices (s = 0 and s = 1). Generalized Fibonacci matrices in certain cases give the usual Fibonacci matrix and the Lucas matrix. Inverse of the matrix U (a,b,s) n is derived. In partial case we get the inverse of the generalized Fibonacci matrix F

The Representations of the Fibonacci and Lucas Matrices

Iranian Journal of Science and Technology, Transactions A: Science, 2019

In this study, a matrix R L is defined by the properties associated with the Pascal matrix, and two closed-form expressions of the matrix function f (R L) = R n L are determined by methods in matrix theory. These expressions satisfy a connection between the integer sequences of the first-second kinds and the Pascal matrices. The matrix R n L is the Fibonacci Lucas matrix, whose entries are the Fibonacci and Lucas numbers. Also, the representations of the Lucas matrix are derived by the matrix function f (R L − 5I) , and various forms of the matrix (R L − 5I) n in terms of a binomial coefficient are studied by methods in number theory. These representations give varied ways to obtain the new Fibonacci-and Lucas-type identities via several properties of the matrices R n L and (R L − 5I) n .

A three by three Pascal matrix representations of the generalized Fibonacci and Lucas sequences

Hacettepe Journal of Mathematics and Statistics

In this study, a matrix R v is defined, and two closed form expressions of the matrix R n v , for an integer n ≥ 1, are evaluated by the matrix functions in matrix theory. These expressions satisfy a connection between the generalized Fibonacci and Lucas numbers with the Pascal matrices. Thus, two representations of the matrix R n v and various forms of matrix (R v +q I) n are studied in terms of the generalized Fibonacci and Lucas numbers and binomial coefficients. By modifying results of 2 × 2 matrix representations given in the references of our study, we give various 3 × 3 matrix representations of the generalized Fibonacci and Lucas sequences. Many combinatorial identities are derived as applications.

On the generating matrices of the Κ-Fibonacci numbers

Proyecciones (Antofagasta), 2013

In this paper we define some tridiagonal matrices depending of a parameter from which we will find the k-Fibonacci numbers. And from the cofactor matrix of one of these matrices we will prove some formulas for the k-Fibonacci numbers differently to the traditional form. Finally, we will study the eigenvalues of these tridiagonal matrices.

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

Abstract and Applied Analysis, 2014

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 and -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 and -circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and -circulant matrices by utilizing the relation between left circulant, and -circulant matrices and circulant matrix, respectively.

Singular case of generalized Fibonacci and Lucas matrices

The notion of the generalized Fibonacci matrix F (a,b,s) n of type s, whose nonzero elements are generalized Fibonacci numbers, is introduced in the paper [23]. Regular case s = 0 is investigated in [23]. In the present article we consider singular case s = −1. Pseudoinverse of the generalized Fibonacci matrix F (a,b,−1) n is derived. Correlations between the matrix F (a,b,−1) n and the Pascal matrices are considered. Some combinatorial identities involving generalized Fibonacci numbers are derived. A class of test matrices for computing the Moore-Penrose inverse is presented in the last section.