Eigenvectors of a 2x2 Hermitian matrix

Question

How do I find what the value of $e_1^\dagger (e_1 +e_2)$ is, where $e_1 ,e_2$ are the eigenvectors of a 2x2 Hermitian matrix? I have absolutely no idea where to even start with this one so any help would be appreciated :)

I know that, for a Hermitian matrix, $H=H^\dagger$, but am not sure if this applies to the eigenvectors as well?

Thanks!

Answer 1

So, expanding the brackets gives: $$e_1^\dagger e_1 +e_1^\dagger e_2$$ This can be written as a dot product: $$\langle e_1 ,e_1\rangle + \langle e_1 ,e_2\rangle $$ As they are an orthonormal basis, $\langle e_1 ,e_1\rangle =1$ and $\langle e_1 ,e_2\rangle =0$. Therefore: $$e_1^\dagger (e_1 +e_2) = 1$$