We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, and $P = I_n - \frac{1}{n} \mathbf{e} \cdot \mathbf{e}^T$. We denote by $\Delta_n$ the set of EDM of order $n$ and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $T$ the application from $\Delta_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to $D$ $$T(D) = -\frac{1}{2} P D P.$$ Let $D \in \Delta_n$. Let $A_1, \ldots, A_n$ be points whose matrix $D$ is the Euclidean distance matrix. We denote by $x_i$ the coordinate vectors of the $A_i$ and $M_A$ the matrix whose columns are the $x_i$ and $C$ the column formed by the $\|x_i\|^2$. Write $D$ as a linear combination of $C\mathbf{e}^T$, $\mathbf{e}C^T$ and $M_A^T \cdot M_A$. Deduce that for every matrix $D$ of $\Delta_n$ we have $T(D) \in \Omega_n$.
We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, and $P = I_n - \frac{1}{n} \mathbf{e} \cdot \mathbf{e}^T$. We denote by $\Delta_n$ the set of EDM of order $n$ and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $T$ the application from $\Delta_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to $D$
$$T(D) = -\frac{1}{2} P D P.$$
Let $D \in \Delta_n$. Let $A_1, \ldots, A_n$ be points whose matrix $D$ is the Euclidean distance matrix. We denote by $x_i$ the coordinate vectors of the $A_i$ and $M_A$ the matrix whose columns are the $x_i$ and $C$ the column formed by the $\|x_i\|^2$. Write $D$ as a linear combination of $C\mathbf{e}^T$, $\mathbf{e}C^T$ and $M_A^T \cdot M_A$. Deduce that for every matrix $D$ of $\Delta_n$ we have $T(D) \in \Omega_n$.