Linear Algebra. Proof with eigenvectors and orthogonal vectors

Question

Let $A$ be an $n\times n$ matrix. Consider $v$ an eigenvector of $A$ associated to the eigenvalue $\lambda$, and $w$ an eigenvector of $A^T$ associated to the eigenvalue $\alpha$. Show that if $\lambda\neq\alpha$ then $v$ and $w$ are orthogonal.

Hello, I have a few doubts about this exercise and would like a way to do this proof. Thanks

Answer 1

Hint : there are two ways to write $w^T A v$ : \begin{align*} w^TAv &= w^T (Av)\\ &= (A^T w)^Tv \end{align*}