Finding pairs of operator-related vectors

[dipole]

Say I have an even-numbered set of vectors, $X = \{x_1, x_2, ...x_{2n}\}$ where there exists some pairing of the vectors such that,
$$x_iA = x'_i \quad \forall i=1..n$$

However, I don't know what the pairing should be. Other than iterating over some norm and finding all pairs of $i$ and $j$ which satisfy $|| x_iA - x_j || = 0$, can anyone think of a faster way of doing it?

 

[FactChecker]

One thing that might speed up the calculations, but would make the algorithm more complicated, would be to test for equality index-by-index and only continue to the next index if the current one is equal. Of course, you should not forget to remove any matches from the list before moving on to the next search.