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)?
Deriving singular values based on column properties
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linear-algebra
matrices
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1@Tapu yes, singular values of $X$ are inverse of square roots of diagonal entries of $D$. Please organize your answer. – 2012-10-10
1 Answers
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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 $X$.