References: edX online course MIT 8.05 Week 6.
A hermitian operator is normal because . A unitary operator is normal because and every operator commutes with its inverse.
Theorem 1 A normal operator remains normal after a similarity transformation with a unitary operator .
Proof: Suppose we transform according to
Thus is normal.
Theorem 2 If is an eigenvector of the normal operator with eigenvalue , it is also an eigenvector of the adjoint with eigenvalue .
Consider the vector
The norm of this vector satisfies