Spectral Theorem for Normal Operator

Theorem

1. With $V$ as a finite-dimensional Complex Inner Product Space
2. Let $T\in \mathcal{L}(V)$ Then, the following are equivalent:
3. $T$ is Unitarily Diagonizable
4. $T$ is Normal

Proof

By Schur’s Decomposition, all operators are unitarily upper trigular. If the operator is upper triangular using the orthonormal bases. it is diagonal $⟺$ it is normal