# Why the sum of cosines between "v" and any vector =1?

1. Feb 12, 2017

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.