what's the advantage of using eigen decomposition for matrix inverse?

Question

today I read an article saying that one of the applications of eigen decomposition is for matrix inverse

$AP= P D$

where P's columns are eigenvectors, and D is a diagonal matrix

$A = PDP^{-1}$

$A^{-1} = PD^{-1}P^{-1}$

I can understand that $D^{-1}$ is easy to solve

but I still need to solve $P^{-1}$, right?

The only thing special about $P$ is that P's columns are orthogonal. Why does this make it easy to inverse ?

Thanks!

Answer 1

The good thing about orthogonal matrices is that $Q^{-1} = Q^T$

Answer 2

I cannot imagine that someone would do this in practice. As a general rule, in practice one should not form the inverse of a matrix. One should solve with a matrix instead.

My suspicion is that inverting the matrix in this way is important when pushing some theoretical results through (for example, when analyzing the inverse power method).