We consider the endomorphism $\Phi$ of $\mathcal{M}_n(\mathbb{R})$ defined by:
$$\forall M \in \mathcal{M}_n(\mathbb{R}), \quad \Phi(M) = PMP$$
where $P = I_n - \frac{1}{n}J$.
1) Show that $\Phi$ is an orthogonal projector in the Euclidean space $(\mathcal{M}_n(\mathbb{R}), (\cdot \mid \cdot))$.
2) Show that $\operatorname{Im}\Phi = \left\{M \in \mathcal{M}_n(\mathbb{R}) \mid MZ = 0 \text{ and } {}^t MZ = 0\right\}$.