Deriving singular values based on column properties

Question

Given a rectangular matrix $X\in\mathbb{R}^{d\times p}$, $d>p$, and a diagonal matrix $D\in\mathbb{R}^{d\times d}$ with positive diagonal entries, and property $$X^TDX=I,$$ with $I\in\mathbb{R}^{p\times p}$ being the identity matrix, could some property be derived on SVD of $X$ (ie, singular values and singular values)?

Answer 1

All right, as I said in comment, your equation can be written as $$Y^tY=I$$ with $Y=\sqrt{D}X$. This shows that the eigenvalues of $Y^TY$, i.e., the singular values of $Y$ are $1$ ($p$ times) and $0$ ($d-p$ times). Now it is possible to find the singular values of $X$.