We consider a matrix $M = (m_{ij}) \in \mathcal{S}_n(\mathbb{R})$ such that for every $(i,j) \in \llbracket 1,n\rrbracket^2$, $m_{ij} \geqslant 0$ and $m_{ii} = 0$. We assume that $\Psi(M)$ has at least one strictly negative eigenvalue.
We seek to prove that there exists a unique symmetric matrix $T_0$ with non-negative eigenvalues that minimizes $\|\Psi(M) - T\|_{\mathcal{M}_n(\mathbb{R})}$ when $T$ ranges over $\mathcal{S}_n^+(\mathbb{R})$.
a) Show that $$\forall Q \in \mathcal{O}_n(\mathbb{R}), \forall A \in \mathcal{M}_n(\mathbb{R}), \quad \|{}^t QAQ\|_{\mathcal{M}_n(\mathbb{R})} = \|A\|_{\mathcal{M}_n(\mathbb{R})}$$
b) Justify the existence of a matrix $Q_0 \in \mathcal{O}_n(\mathbb{R})$ such that the matrix ${}^t Q_0 \Psi(M) Q_0$ is diagonal.
c) Show that a necessary condition for $\|\Psi(M) - T_0\|_{\mathcal{M}_n(\mathbb{R})}$ to minimize $\|\Psi(M) - T\|_{\mathcal{M}_n(\mathbb{R})}$ when $T$ ranges over $\mathcal{S}_n^+(\mathbb{R})$ is that the matrix ${}^t Q_0 T_0 Q_0$ is diagonal.
d) Prove the existence and uniqueness of the matrix $T_0$ sought.