Ok so I am trying to show that if A is normal and has real eigenvalues, then A is hermitian.
It was suggested that I try using the spectral theorem. So if we assume A is normal and has real eigenvalues, then we can write $A = UDU^*$, where $U$ is unitary and $D$ is a diagonal matrix with real eigenvalues as its entries. But I am not sure of the next step?
Note $D=D^*$ because $D$ is real and diagonal, so
$$A^*=(UDU^*)^*=UD^*U^*=UDU^*=A.$$
If $A$ is normal with real eigenvalues, then, as you say, $A=UDU^{\star}$ where $U$ is unitary and $D$ is a diagonal matrix with real values on the diagonal. So $D^{\star}=D$, which implies $$ A^{\star}=(UDU^{\star})^{\star}=U^{\star\star}D^{\star}U^{\star}=UDU^{\star}=A. $$
