Question

The Fibonacci Numbers form a sequence F with F0 = 0 and F1 = 1 satisfying the recurrence relationship Fn+2 = Fn + Fn+1. The first few terms are therefore: F = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... (a) (2 marks) Prove that the fibonacci numbers satisfy the following matrix equation: [Fn+1 Fn] = [1 1 1 0]^n [1 0] (b) (3 marks) Compute the eigenvalues and corresponding eigenvectors for the matrix: A = [1 1 1 0] Give your answers as exact values. (c) (2 marks) Using your answers from part (b) of this question, diagonalise the matrix A. That is, find matrices P and D with D diagonal such that A = PDP^-1. (d) (2 marks) Hence obtain a closed-form expression for Fn using the diagonalisation found in part (c) and the equation (1).

          The Fibonacci Numbers form a sequence F with F0 = 0 and F1 = 1 satisfying the recurrence relationship Fn+2 = Fn + Fn+1. The first few terms are therefore:
F = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(a) (2 marks) Prove that the fibonacci numbers satisfy the following matrix equation:
[Fn+1 Fn] = [1 1 1 0]^n [1 0]
(b) (3 marks) Compute the eigenvalues and corresponding eigenvectors for the matrix:
A = [1 1 1 0]
Give your answers as exact values.
(c) (2 marks) Using your answers from part (b) of this question, diagonalise the matrix A. That is, find matrices P and D with D diagonal such that A = PDP^-1.
(d) (2 marks) Hence obtain a closed-form expression for Fn using the diagonalisation found in part (c) and the equation (1).

The Fibonacci Numbers form a sequence F with F0 = 0 and F1 = 1 satisfying the recurrence relationship Fn+2 = Fn + Fn+1. The first few terms are therefore:
F = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(a) (2 marks) Prove that the fibonacci numbers satisfy the following matrix equation:
[Fn+1 Fn] = [1 1 1 0]^n [1 0]
(b) (3 marks) Compute the eigenvalues and corresponding eigenvectors for the matrix:
A = [1 1 1 0]
Give your answers as exact values.
(c) (2 marks) Using your answers from part (b) of this question, diagonalise the matrix A. That is, find matrices P and D with D diagonal such that A = PDP^-1.
(d) (2 marks) Hence obtain a closed-form expression for Fn using the diagonalisation found in part (c) and the equation (1).

Added by Robert M.

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