We denote by $D$ the diagonal matrix of size $n$:
$$D = \operatorname{Diag}\left((1 - \alpha_j^2)_{1 \leq j \leq n}\right)$$
and $V \in \mathcal{M}_n(\mathbf{R})$ the matrix such that for every $j \in \llbracket 1, n \rrbracket$, the $j$-th column of $V$ is $V_j = f_j(S^\top) U$. Show that
$$J(p) = V D V^\top.$$