Let $A$ and $B$ be commuting $n \times n$ Hermitian matrices. Prove that $A$ and $B$ can be simultaneously unitary diagonalized. Also Prove that $AB = 0 = BA$, where $0$ is the zero matrix, implies that rank ($A + B$) = rank ($A$) + rank($B$).
Thanks in advance!
This is what I have so far. We know that $AB$ and $BA$ have a common eigenvector. The goal is to find a unitary matrix $S$ so that $$D_A = S^* A S \quad \text{and} \quad D_B = S^* B S$$ where $D_A$ and $D_B$ are diagonal matrices.