Prove that a normal operator with real eigenvalues is self-adjoint Seems like a simple proof, but I can't seem to get it. My attempt: We know that a normal operator can be diagonalized, and has a complete orthonormal set of eigenvectors. Let A be normal. Then A= UDU* for some diagonal matrix D and unitary U. Also, A*=U*D*U Since D is the diagonal matrix of the eigen values of A, D is real, and thus D=D*. Thus D=U*AU = UA*U*. Then, I just get stuck on A=U²A*(U*)².