Does each distinct eigenvalue of a matrix correspond to a unique eigenspace?

Question

I'm wondering if each distinct solution of $λ$ for $Ax=λx$ corresponds to a unique eigenspace for $x$. That is, is it possible to have two distinct solutions to $Ax=λx$ in terms of lambda ($λ$), that happen to have the same eigenspace of associated vectors?

Answer 1

Of course not:
If $\lambda x=Ax=\lambda' x\,$ then $(\lambda-\lambda')x=0$, hence either $x=0$ or $\lambda=\lambda'$.