HW #17 (Due Wed. Dec. 6 )

1. Find the dimensions and bases for all eigenspaces of the following operators

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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

\begin{displaymath}\left(\begin{array}{c}F_{n-1}\\ F_n\end{array}\right) =
A=\left(\begin{array}{cc}0&1\\ 1&1\end{array}\right)\ .\end{displaymath}

3. Suppose that the matrices A and B are similar. So there exist an invertible matrix P such that $A=P^{-1}\cdot B\cdot P$. Prove that $A^n=P^{-1}\cdot B^n\cdot P$ for any integer n.

4. Diagonalize the matrix $A=\left(\begin{array}{cc}0&1\\ 1&1\end{array}\right)$ from the problem 2. This means find an invertible matrix P and a diagonal matrix B such that $A=P^{-1}\cdot B\cdot P$.

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.