We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, $P = I_n - \frac{1}{n}\mathbf{e}\cdot\mathbf{e}^T$, and $\Delta_n$ the set of EDM of order $n$.
Show that every non-zero symmetric matrix with non-negative coefficients and zero diagonal, having a unique strictly positive eigenvalue with eigenspace of dimension 1 and eigenvector $\mathbf{e}$, is EDM.