# Fibonacci rule for eigenvalues

I'm having difficulty with the following, problem 5.3.7 from Gilbert Strang's "Linear Algebra and its Application".

The numbers $\lambda_1^k$ and $\lambda_2^k$ satisfy the Fibonacci rule $F_{k+2}=F_{k+1} + F_k$ :

$\lambda_1^{k+2}=\lambda_1^{k+1} + \lambda_1^k$ and $\lambda_2^{k+2}=\lambda_2^{k+1} + \lambda_2^k$

Prove this by using the original equation for the $\lambda's$ (Multiply it by $\lambda^k$)

Then any combination of $\lambda_1^k$ and $\lambda_2^k$ satisfies the rule.

The combination $F_k=(\lambda_1^k-\lambda_2^k)/(\lambda_1-\lambda_2)$ gives the right start of $F_0=0$ and $F_1=1$.

I'm not sure what "the original equation for the $\lambda's$" is, or what I'm supposed to prove. Can someone please help?

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What is "the original equation for the $\lambda'$s"? Where did you encounter this problem, and in what context? The more you can tell us about that, the more likely we'll be able to help you. –  Cameron Buie Nov 11 '12 at 16:26
That is the part that also confuses me. It is a problem of the book "linear algebra and its application, 5.3.7", and I'm not sure what the 'original equation' means, also I can't find any 'equation for lambda' in previous pages... –  existence Nov 11 '12 at 16:30
Anyway, I can't understand what is the main point of this problem. What should I prove? The above two equations for lambdas? By using the Fibonacci formula? –  existence Nov 11 '12 at 16:32
Who would the book's author(s) be? Have you stated the entire problem? –  Cameron Buie Nov 11 '12 at 16:33
The author is Gilbert Strang. –  existence Nov 11 '12 at 16:38

It looks to me like $\lambda_1$ and $\lambda_2$ were defined earlier, probably as eigenvalues of the matrix $\left( \begin {matrix} 1&1\\1&0 \end {matrix} \right)$ so $\lambda_1=\frac 12(1+\sqrt 5)$ and $\lambda_2=\frac 12(1-\sqrt 5)$. You are now supposed to prove the sentence just under "Fibonacci Rule".

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The proof first takes the $\lambda s$ and converts these into two different solutions to the fibonacci equation. Then you will need to show that if you have these different solutions, any linear combination of them - i.e. $a\lambda_1^n+b\lambda_2^n$ will also fit the fibonacci rule. The third part of the proof is to find the combination which gives the correct first two values, solving for $a$ and $b$. Once this is done, you are done, because the first two values determine all the others by repeated application of the rule. –  Mark Bennet Nov 11 '12 at 17:04