The way I have proved the Eigen values of Hermitian matrices are real like this:
I considered $H$ is a hermitian matrix.
Operator applied in ket space: $\left<\psi|H | \psi\right>= \lambda$
Operator applied in bra space: $\left<\psi|H | \psi\right> = \lambda^\star \ $
There is no state in the right hand side because of the normalization.
If we subtract the two equations we get $\lambda = \lambda ^*$ Therefore the eigenvalues are real
Have I done in the right way?