Let $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and let $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ be a symmetric matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Show that $S$ is symplectic.
One may consider a basis of eigenvectors of the endomorphism $s$ of $\mathbb { R } ^ { n }$ canonically associated with $S$, and show that $s$ is a symplectic endomorphism of the standard space $(\mathbb { R } ^ { n } , b _ { s })$.