Let $M$ be a symmetric matrix of $\mathcal{M}_n(\mathbf{R})$. Show that if $M$ is positive, then there exists $B \in \mathcal{M}_n(\mathbf{R})$ such that $M = B^T \cdot B$. Deduce that if $M$ is no longer assumed to be positive, but admits a unique strictly positive eigenvalue $\lambda$ with eigenspace of dimension 1 and unit eigenvector $u$, then there exists $B \in \mathcal{M}_n(\mathbf{R})$ such that $M = \lambda u \cdot u^T - B^T \cdot B$.