This is elementary, yet useful and interesting.
Statement: All eigenvalues of a Hermitian matrix are real.
If the matrix is Hermitian, by definition,
Let be an eignevalue of , and be the corresponding eigenvector. Let
Since is an vector, is just a number (real or complex).
Now, it follows from (1) that .
Any complex number which is equal to its conjugate must have the imaginary part equal to zero. Therefore, must be real, which implies must also be real. Since the sum is also real, it follows the must be real.