MEINKE, ASHLEY MARIE, M.S. AUGUST 2011 MATHEMATICS
FIBONACCI NUMBERS AND ASSOCIATED MATRICES (43 pp.)
Director of Thesis: Aloysius Bathi Kasturiarachi
In this thesis, we approach the study of Fibonacci numbers using the theory of matrices. Fibonacci numbers are widely studied and the formulas to derive them are well-known. Such formulas include Binet's formula and Cassini's formula. We use the theory of diagonalizing a matrix and examine eigenvalues of certain 2 x 2 generating matrices to derive Binet-type formulas for the Lucas, generalized Fibonacci and generalized weighted Fibonacci numbers. We derive Cassini-type formulas for the Lucas, generalized Fibonacci and generalized weighted Fibonacci numbers by computing determinants of certain matrices. We extend these results to Tribonacci and generalized Tribonacci numbers using a similar 3 x 3 matrix approach. In all cases, we do a thorough analysis of the recursive sequences versus the derived Binet-type formulas. Throughout the thesis, we make notes of important historical information and examine Fibonacci's life and mathematical work. We also discuss the presence of these numbers even before Fibonacci's time by looking at the role that Indian mathematics played in the “so-called” Fibonacci numbers. In addition, we provide a sampling of some of the interesting properties that Fibonacci numbers possess.