Prove that idempotency is a necessary and sufficient condition for a densiy matrix to be N-representable
A density matrix $D$ is positive semidefinite, hermitian, and has trace one.
Because of hermiticity we may assume that it is diagonal.
Let's denote the eigenvalues with $\lambda_i$ .
Because it is positive semidefinite, we have $0 \leq \lambda_i \leq 1$.

The matrix $D^2$ has eigenvalues $\lambda_i^2$. Since it is idempotent ($D = D^2$) and has trace one, we may write:
$$\sum_i \lambda_i = \sum_i \lambda_i^2 = 1$$
The only possibility for this to be true is, if there is one and only one $j$
for which $\lambda_j = 1$.

So $D$ has one $1$ in the diagonal at the $j$-th column and all other values in the matrix are zero.
You can easily represent this matrix with:
$ e_j e_j^T$. ($e_j$ being the $j$-th unit vector.)

If you transform back to you original nondiagonal form, you obtain your $\Psi$.