1. ## Eigenvalue problem

This is my attempt at a solution:

A^2(x) = A(A(x))
A^2(x) = lambda^2(x)

Therefore x is an eigenvector of A^2 with eigenvalue lambda^2. The statement for general N>0 follows in the same way.

Am I missing anything?

2. Originally Posted by temaire

This is my attempt at a solution:

A^2(x) = A(A(x))
A^2(x) = lambda^2(x)

Therefore x is an eigenvector of A^2 with eigenvalue lambda^2. The statement for general N>0 follows in the same way.

Am I missing anything?

Not really: now just round up the proof doing some induction.

Tonio

3. Originally Posted by tonio
Not really: now just round up the proof doing some induction.

Tonio
I haven't learned what induction is. Or maybe its a word for something I already know. What do you mean by it?

4. A proof by induction works like this: first show your statement holds for $n=1$ or $0$ or the first $n$ for which it is true, then assume it to be true for $n=m$ (this is your induction hypothesis) and then show it to be true for $n=m+1$ using your hypothesis. if $\lambda$ is an eigenvalue of $A$, $Ax=\lambda x$ and $A^2x=A\lambda x =\lambda Ax=\lambda^2 x$ so $\lambda^n$ is an eigenvalue of $A^n$ for $n=2$. suppose it is true for $n=m$,then $A^{m+1}x=AA^mx=A\lambda^mx=\lambda^mAx=\lambda^{m+ 1}$
it is essentially what you said but a a bit more rigorous
induction is a really useful tool, you should practice using it