Spectral theorem

The spectral theorem states that if an operator $A \in \mathcal L(V)$ on a Hilbert space $V$ is normal (i.e. $[A,A\adj] = 0$), its eigenvectors form an orthonormal basis of $V$.

A normal operator $A$ can be diagonalized by a unitary transformation

Where $\Lambda$ is a diagonal operator with the eigenvalues of $A$ as its entries (in the finite-dimensional case).