For every integer $j \in \llbracket 1, n \rrbracket$, we define the matrices $$B_j = S - \alpha_j I_n \quad \text{and} \quad C_j = I_n - \alpha_j S$$ Prove that $$J(p) = \sum_{j=1}^{n} f_j(S)^\top \left(C_j^\top C_j - B_j^\top B_j\right) f_j(S)$$