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 $n$.
Let $D$ be a non-zero EDM of order $n$. Let $\lambda_1, \ldots, \lambda_n$ be its eigenvalues, ordered in increasing order. Show
$$\lambda_{n-1} \leqslant 0$$
and deduce that $D$ has exactly one strictly positive eigenvalue.