SVD of $H^TXH$ with SVD of $X$ known

Question

let $U D U^T$ be the singular value decomposition of $X \in \mathbb{R}^{d \times d}$, a symmetric positive definite matrix, and let $H \in \mathbb{R}^{d \times n}$ be a rank-$d$ rectangular matrix. Can we say something about the SVD of $H^TXH$ with this information?

Answer 1

Propably the best thing, you can do, is to calculate the QR-Decomposition of $H$ and evaluate $RXR^T$ to calculate it's SVD ($V\Sigma V^T$). Then you multiply $QV$. Since $H$ is neither orthogonal nor anything useful, you have scaling and other stuff in there, that will destroy every property of the SVD of $X$.