Say there is a 3x3 symmetric matrix such as the following: $$A=\begin{bmatrix} a & b & a \\b & b & b \\ a & b & a \end{bmatrix}$$
By being symmetric:$$AA^T=A^TA=A\cdot A =A^2$$
$$\text{Which means that the left & right singular vectors are the same } (\text{eigenvectors of } AA^T \text{ are the same as the eigenvectors of } A^TA \text {)}$$
$$\text{So,one can say: } A=UΣV^T=UΣU^T \\ \text{But that is not the case,there is a different sign at one of the eigenvectors of }V^T \text{.}$$
The question is,why does that happen?Is it supposed to happen? Turns out,if that vector that was different by a sign is computed manually,using the relation between the left & right singular vectors,the correct sign is given and SVD is complete.
Any ideas?