Let $A$ be a square matrix over $\mathbb C$, and let $A^T$ denote its transpose.
It is not hard to see that $A$ and $A^T$ have the same set of eigenvalues, so given $Ax=\lambda x$ for some vector $x\in V$ and eigenvalue $\lambda\in\mathbb C$, we know that there must always be some other $y\in V$ such that also $$A^T y=\lambda y.$$
We also know that, if $Ax=\lambda x$ and $A^T y=\mu y$ with $\lambda\neq \mu$, then $\langle y^*,x\rangle=0$, where $y^*$ denotes the vector whose elements are complex conjugate of those of $y$, as follows from $$\langle y^*,Ax\rangle=\lambda \langle y^*,x\rangle=\mu\langle y^*,x\rangle.$$
The same argument, however, does not provide any information for the case $\mu=\lambda$. Is there relation holding in general for such a case?
More precisely, given $Ax=\lambda x$ and $A^T y=\lambda y$, is there any general relation between $x$ and $y$?