Let $x$ and $y$ be unit vectors in the Euclidean norm. Define $s(\lambda) = |y^Hx|$, where $Ax=\lambda x$ and $ y^HA=\lambda y^H$. Here $\lambda$ is a simple eigenvalue (an eigenvalue with algebraic multiplicity $1$). I would like to prove $s(\lambda) \leq 1$ and $s(\lambda)\neq 0$. I tried the following and gave up after that.
$Ax=\lambda x $
Premultiply by $y^H$
$\begin{align*} y^HAx &= \lambda y^Hx \\ y^HAx &= \lambda y^H I x \end{align*}$
Taking norm on both sides,
$ \begin{align*} \frac{1}{\lambda} \|y^HAx\| &= \|y^H I x \| \\ \frac{1}{\lambda} \|y^HAx\| &= |y^Hx| \end{align*}$
Can anyone help me how to proceed after this? (Similar question, with a matrix free proof here:Proving that two right and left eigenvectors are not orthogonal.)