How to generate unique SVD decomposition problems?

Question

As far as I understand, if I first generate 2 orthogonal matrices U and V, then generate a random diagonal matrix with random eigenvalues, I end up with a matrix, that might possibly have a different SVD decomposition. All of my matrices are 3x3. Is there some kind of an extra condition, that I can enforce to assure, that the values used for the problem generation are the one and only possible solution?

Answer 1

Uniqueness of SVD is not guaranteed in general. Take for example identity matrix.

We can construct a rotation matrix $\mathbf R$ for $\theta$ radians so that :

$${\mathbf I}_n = {\mathbf R} {\mathbf I}_n {\mathbf R}^T$$

for any value of $\theta$. In particular this happens if the eigenspace is non-simple.

If we have a projection instead of identity we will get even more degrees of freedom since we can rotate arbitrarily within the subspaces that have singular value 0.