Characterization of Fibonacci Matrices via Tridiagonal Symmetric Toeplitz Structures

Abstract

This study investigates the properties of Fibonacci matrices, with a particular focus on the tridiagonal symmetric Toeplitz matrix \( S_4(x,y) \), whose entries are associated with Fibonacci numbers \( F_n \) and Lucas numbers \( L_n \). It extends Filipponi's foundational analysis by investigating distinct cases of the ordered pairs  \( (x,y) \), determining the precise conditions under which \( S_4(x,y) \) qualifies as a Fibonacci matrix and identifies the specific conditions under which \( S_4(x,y) \) can be classified as a Fibonacci matrix. The analysis derives closed-form expressions for the entries of \( S_4^n(x,y) \) for distinct ordered pairs, including \( (x,y)=(F_{s+1},F_s) \), \(  (x,y)=(F_{-s},F_{-(s+1)}) \) and \( (x,y)=(F_{-(s+1)},F_{-s}) \). These findings extend the theoretical framework of Fibonacci matrices and offer new insights into their structural properties.