Is there an intuition behind the quadratic form of SPD (symmetric and positive definite) matrices:
$$(S, \Sigma) \in \mathcal{S}_{++}\times\mathcal{S}_{++} \mapsto S\Sigma S$$ where $S$ and $\Sigma$ are positive definite symmetric $d\times d$ matrices.
I thought it measures the similarty of the rows/ columns of $S$ transformed by the eigenvectors of $\Sigma$ since the $i,j-th$ entry of $S\Sigma S$ is $s_i^T \Sigma s_j$ where $s_i$ is the i-th column of S and using eigendecomposition $\Sigma=U^TDU$ one gets $s_i^T \Sigma s_j= (Us_i)^TD (Us_j)$.
Is there any other interpretation / intution behind the quadratic form of SPD matrices?