# Reshaping $U$ and $V^\dagger$ matrices resulting from an SVD into rank-3 tensors

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 $$S$$.

Using the tensor network notation I want to achieve the following:

Or maybe there is some direct transition from the $$M$$ matrix to the final picture, which I just didn't see?

If you still want $$U$$ and $$V$$ to be left/right normalized, then I don't think it is generally possible to split in your way. The singular value decomposition is unique up to phases if the singular values are not degenerate. Therefore, you are only allowed to change $$S$$ up to phases. According to your splitting, this means that $$\mathrm{diag}(e^{i\theta}) S = S_0 \otimes S_1$$. However, this is not possible if $$S$$ is entangled (that is, it has more than one nonzero singular value) regarded as a reshaped matrix of dimensions $$4\times 9$$, which is in general the case.