$\Rightarrow$To start with, according to the similarity theorem, they have the same eigenvalues so is the multiplicity.
$\Leftarrow$If two matrices have the same eigenvalues and multiplicity, that implies they are similar to the same diagonalizable matrix $D$ and by equivalent relation, they are similar to each other.
However for the second part, since we are not sure the dim($E_{\lambda n})=$multiplicity for $\lambda_n$, how are we making sure that the two matrices are diagonalizable. If diagonalizability is not assured, my second part is wrong.
Could you offer me any hint?
We are assuming that the matrices are diagonalizable. So, instead, you should show that they are similar to the same diagonal matrix $D,$ and conclude from there.