Proof about Diagonalization of A The question asks WHY is it true that $$A^{n} = PD^{n}P^{-1}$$
I can never do proper proving in algebra; what I almost know for sure is that a proof by induction is the way to go here. But how do you go about proving with induction, whenever I think of starting a proof, I just end up blocked at the initial hypothesis. How do you carry on?
Moreover, is it legitimate to say that for $$A^{1} = PD^{1}P^{-1}$$
It holds true simply because of the hypothesis/theorems with started out with?
Thanks

initially when you suppose that $A^1=PD^1P^{-1}$ is because $A$ is diagonalizable. Then using induction method, you suppose that is true for the case $n$, then you have that prove that this is true for $n+1$:
$A^{n+1}=A^nA^1=(PD^nP^{-1})(PD^1P^{-1}),$
in the last equality we use the induction hypothesis, finally multiply this product and obtain the answer.
