Let's say, that we have a $6 \times 6$ matrix $M$. By conducting an SVD of $M$ we obtain $USV^\dagger$ matrices, where $U$ is of size $6 \times 6$ and is left-normalized, $S$ is diagonal with 6 singular values on the diagonal, and $V^\dagger$ is right-normalized, also of size $6 \times 6$.
Now, I want to reshape $U$ into a rank-3 tensor of size $6 \times 2 \times 3$, and similarly $V^\dagger$ into a $2 \times 3 \times 6$ tensor. My question is, is it possible to associate two diagonal matrices $S_0$ (of size $2 \times 2$) and $S_1$ (of size $3 \times 3$) to each of the newly obtained dimensions of $U$ and $V^\dagger$? By doing that, it would be best to somehow inference values of $S_0$ and $S_1$ from the initial $S$.
Or maybe there is some direct transition from the $M$ matrix to the final picture, which I just didn't see?