1. Find the dimensions and bases for all eigenspaces of the following operators
The purpose of the following problems 2-6 is to derive a closed formula for Fibonacci numbers Fn (F0=1, F1=1, Fn= Fn-1+Fn-2 for all integers n>2). The ``closed formula" means not a recurrent formula as above which expresses Fn in terms of previous numbers, but rather a formula which expresses Fn in terms of n only.
2. Prove that
3. Suppose that the matrices A and B are similar. So there exist an invertible matrix P such that . Prove that for any integer n.
4. Diagonalize the matrix from the problem 2. This means find an invertible matrix P and a diagonal matrix B such that .
5. (a) Find the matrix Bn for B from the previous problem.
(b) Using the problem 3 find An for A from the previous problem.
6. Find the closed formula for Fibonacci number Fn.