Eigenvector and orthogonal matrix

Question

If $D$ is a diagonal matrix and $P$ is an orthogonal matrix where $P^{-1} = P^T$.

If $y= PDP^T$ , why is $y^{100} = P D^{100} P^T$ and not $P^{100} D^{100} {(P^T)}^{100}$ ?

Answer 1

Well, let's take a look at $y^2 = PDP^T \cdot PDP^T$. The $P^T$ and $P$ in the middle cancel out because $P^T$ is the inverse of $P$. That means this can be simplified to $y^2 = PD^2P^T$. Do you see the pattern?