It is proven that $A$ and $A^T$ have the same eigenvalues. I want to study what stands for eigenvectors. Let me make a try. Given:
$$Ax=\lambda x$$ we know that $x\in C(A)$ for $\lambda \neq 0$. Suppose that for $A^T$ we have the same eigenvectors $x$:
$$A^Tx=\lambda x$$ but now we have that $x\in C(A^T)$. Based on this, eigenvector's $x$ belong both in column and row space which is impossible. So, $A$ and $A^T$ have different eigenvectors.
Am I right about this deduction? In any case, could you please suggest a different way if possible?
PS: After @G Tony Jacobs comments I made some changes hopping that I have less mistakes.