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?



Jacobian of changing of variables to singular value decomposition
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,](https://en.wikipedia.org/wiki/Wishart_distribution)
$$J=\prod\_{i<j}|\sigma\_i-\sigma\_j|\prod\_k\sigma\_k^{(n-m)/2}.$$


