linear algebra - How to prove two diagonalizable matrices are similar iff they have same eigenvalue with same multiplicity.

$\Rightarrow$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?