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**1. The problem statement, all variables and given/known data**

Given that matrix, A can be decomposed using SVD (Singular Value Decomposition) into ##A=USV^T##`, why does always the sum of the square of cosines between v` vectors and any other column vector q representation of arbitrarily column vector Q vector sum up to 1?

**2. Relevant equations**

##A=USV^T##. ##Q=USq##

**3. The attempt at a solution**

I tried a simple 2x2 matrix but even with this simple matrix, the calculation goes missy. In addition, I seek a rigorous proof.