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 $M = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$. It has been shown that ${}^t UU = -\frac{1}{2}\Phi(M)$. Deduce, for every pair $(i,j) \in \llbracket 1,n\rrbracket^2$, an expression for the inner product $\langle U_i, U_j\rangle = {}^t U_i U_j$ as a function of $$\alpha_{ij} = -\frac{1}{n}\left(S(M)_i + S(M)_j\right) + \frac{1}{n^2}\sigma(M)$$ and of $m_{ij}$ (Torgerson relation).
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 $M = \left(\|U_i - U_j\|^2\right)_{(i,j) \in \llbracket 1,n\rrbracket^2} \in \mathcal{S}_n(\mathbb{R})$. It has been shown that ${}^t UU = -\frac{1}{2}\Phi(M)$.
Deduce, for every pair $(i,j) \in \llbracket 1,n\rrbracket^2$, an expression for the inner product $\langle U_i, U_j\rangle = {}^t U_i U_j$ as a function of
$$\alpha_{ij} = -\frac{1}{n}\left(S(M)_i + S(M)_j\right) + \frac{1}{n^2}\sigma(M)$$
and of $m_{ij}$ (Torgerson relation).