I have heard it often that singular values for the SVD are "uniquely determined by the matrix $A$ in $A=U\Sigma V^*$
Wikipedia says it, and so do other sources.
Now, I want to check how I am applying the aforementioned fact. I'm solving this problem:
If A and B have the same singular values, are they unitarily equivalent?
My answer to this is "yes", because singular values are unique for each matrix. So not only are they unitarily equivalent, but they are the same matrix.
However, I see other answers to this question as this is false? (For example here in Exercise 4.4).
What is generally meant by "uniqueness of singular values if it does not meant that each set of singular values corresponds to a different matrix?