# Jacobian of changing of variables to singular value decomposition

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?

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
• 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. Commented Dec 3, 2021 at 1:18