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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0I assumed the diagonal matrix was supposed to be called $D$, and editied your question accordingly. – 2012-10-10
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0It is tempting to multiply the equation by a pseudo-inverse of $X$ on the right, and its transpose on the left. In the general case, the projection on the range of $X$ will not commute with $D$, however, so this does not settle the question. But it might be a beginning. – 2012-10-10
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1This might be helpful...(but I am not sure), your equation implies $Y^tY=I$ with $Y=\sqrt{D}X$. So, we get the singular values of $Y$. Can we get the singular values from $X$, then? – 2012-10-10
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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