Why is unitary diagonalization works?

Question

I've been told that if you take an Hermitian matrix, find it's eigenvectors, normalize them and write them as columns of a matrix, $P$ then:

$$P^{-1}AP = D$$

Where (Magically) $D = \text{Diag}(\lambda_1,\ldots,\lambda_n)$ ($\lambda_i$ is an eigenvalue of $A$).

So I really want to use this algorithm but first I wish to understand why does this magic works?

Answer 1

By Schur decomposition, every square matrix $A$ is unitarily similar to an upper triangular matrix $U$. In particular if $A$ is Hermitian, then $$U^* = (Q^* A Q)^* = Q^* A^* Q = Q^* A Q = U$$ which shows that $U$ is a diagonal matrix.