Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix. We denote $\mathrm{S} = Z^{T}Z$. Show that there exists a decreasing family $(\lambda_{i})_{1 \leqslant i \leqslant d}$ of strictly positive real numbers and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Su_{i} = \lambda_{i} u_{i}$ for all $1 \leqslant i \leqslant d$.