What is the coefficient matrix of the Fibonacci sequence equation?
The Fibonacci sequence is a sequence of numbers in which each number is the sum of the two preceding ones, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, ... The Fibonacci sequence can be represented by the linear homogeneous recurrence relation: F(n) = F(n-1) + F(n-2) Where F(n) represents the n-th number in the sequence. To find the coefficient matrix for this equation, we can write it in matrix form as: [F(n), F(n-1)] = [F(n-1), F(n-2)] * [1, 1; 1, 0] The matrix [1, 1; 1, 0] is the coefficient matrix for this equation. It represents the transformation that maps the n-1th and n-2nd numbers in the sequence to the nth number. This matrix is called the Fibonacci matrix and it plays a crucial role in various areas of mathematics and science, including number theory, graph theory, and the study of dynamical systems.