Let $\mu_i \in \mathbb{R}_{\ge0}$, $i=1,...,n$, and $D=(\delta_{ij}\mu_i)\in\text{M}(n,\mathbb{R})$.
$D$ itself is surely positve semidefinite, this can be proved via prinipal minors.
But why is $S^tDS$ also positive semidefinite for a matrix $S$ with real entries? Is there a proof using principal minors?