Given a normal complex matrix $A$ and knowing that $A = U D U^\dagger$, how can we rewrite the spectral decomposition such that $$U D U^\dagger = \sum_i \lambda_i u_i u_i^\dagger$$ where $\lambda_i$ are the eigenvalues and $D$ is the matrix of eigenvalues?
Rewriting it likely is trivial, however I have not found a nice way to show this myself or online.