I find this problem when reading a paper. The author seemed to think it is trivial so did not list it as a lemma or something. The question is :

$\tilde{u}$ is a random unit vector in $R^D$, $u$ is a fixed unit vector in $R^D$.

$\tilde{W}$ is a random projection matrix down to $R^k$ where $k \leq D$, $W$ is a fixed projection matrix down to $R^k$.

Now set $v_1 = \tilde{W}^T u$, $v_2 = W^T \tilde{u}$, and how to prove that $\|v_1\|$ and $\|v_2\|$ have the same distribution?


My proof(may be wrong):

For all $\tilde{W},$ there exists a matrix $U$ (specifically a rotation and a shift) such that $ W = U \tilde{W}$, then one can get: $$ v_1 = \tilde{W}^T u $$ $$ v_2 = \tilde{W}^T (U^T \tilde{u}) $$ Therefore, for every event $\{\tilde{W} = W_0\} \in \Omega_{\tilde{W}}$, there exists an event $\{\tilde{u} = u_0\} \in \Omega_{\tilde{u}}$, such that $U^Tu_0 = u$, which means $v_1 = v_2$.

Moreover, we can find a bijection $f : \Omega_{\tilde{W}} \rightarrow \Omega_{\tilde{u}}$ that satisfies what mentioned above, which can lead to an identical bijection $g : \Omega_{v_1} \rightarrow \Omega_{v_2}$. Then we have the desired result.


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