Let $U_1, U_2, \cdots, U_n$, $n$ elements of $\mathbb{R}^p$ satisfying $\sum_{i=1}^n U_i = 0$. We define the matrix of squared mutual distances $M = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$. We denote $U$ the matrix of $\mathcal{M}_{p,n}(\mathbb{R})$ having as column vectors the elements $U_1, U_2, \cdots, U_n$.
Show that ${}^t UU = -\frac{1}{2}\Phi(M)$.