Let $M = (m_{ij}) \in \mathcal{S}_n(\mathbb{R})$. We set
$$S(M) = MZ = \left(\begin{array}{c}\sum_{i=1}^n m_{1i}\\ \vdots \\ \sum_{i=1}^n m_{ni}\end{array}\right) = \left(\begin{array}{c}S(M)_1\\ \vdots \\ S(M)_n\end{array}\right) \quad \text{and} \quad \sigma(M) = \langle Z, S(M)\rangle$$
Show that
$$\Phi(M) = M - \frac{1}{n}\left(S(M)^t Z + Z^t S(M)\right) + \frac{\sigma(M)}{n^2}J$$