I'm studying the SVD, but I'm confused about the term "up to complex signs"
If A is square and the singular values are distinct, the left and right singular vectors are uniquely determined up to complex signs
For example, I know that the term can applied to 2 - i and 2 + i. But can it be applied to 2 and -2?
What exactly does "up to complex signs" mean? Can negation to not only complex part but also entire number be allowed?
Up to complex signs seems to me like it would mean some property holds for any complex numbers of the same modulus, as you could interpret complex sign as simply a rotation in the complex plane. Thus the complex sign of a number $z\in\mathbb{C}\setminus\{0\}$ would simply be $\frac{z}{\lvert z\rvert}$. Thus, if your property holds for $z=re^{i\theta}$, then it holds for any number $re^{i(\theta+\phi)}$ for arbitrary $\phi$.