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It is well known that changing variables from a symmetric matrix to its eigenvalue decomposition involves a Jacobian which is just the Vandermonde determinant of the eigenvalues.

Now suppose I have a rectangular matrix and I want to change variables to its singular value decomposition. What is the Jacobian of this transformation?

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For an $m\times n$ real matrix $A=U\Sigma V^t$ with $m\leq n$, diagonal matrix of singular values $\Sigma={\rm diag}\,(\sqrt\sigma_1,\sqrt\sigma_2,\ldots\sqrt\sigma_m)$, orthonormal left and right eigenvector matrices $UU^t=VV^t=\mathbb{1}$, the Jacobian $J$ in the measure $dA=JdUdV\prod_{i=1}^m\sigma_i$ follows from the Wishart distribution, $$J=\prod_{i<j}|\sigma_i-\sigma_j|\prod_k\sigma_k^{(n-m)/2}.$$

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  • $\begingroup$ Is there anything like $\frac{1}{2^mm!}$ in $J$ to account for the fact that by permuting $\sigma_i$ and columns of $U,V$ simultaneously can lead to the same $A$? Also, by changing the signs of each column of $U,V$ simultaneously can also lead to the same $A$? Basically this change of variable is not unique unless we restrict the order of $\sigma_i$ and the sign of the first column of $U$ and $V$ to be positive. $\endgroup$ Commented Dec 3, 2021 at 1:18

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