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Suppose I want to compute the expectation of $\|x'O\|_2$ where $O$ is a random matrix having the uniform distribution on the set of $n \times r$ $(r \leq n)$ semi-orthogonal matrices (i.e., $O'O = I_r$.); $x \in \mathbb{R}^n$ is fixed.

My intuition tells me that, ${\rm E}(\|x'O\|_2) = \sqrt{\frac{r}{n}}\|x\|_2$ (where the expectation is taken wrt the distribution for $O$) but I'm having a difficult time proving this. How would I go about this? Are there any references I am missing?

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  • $\begingroup$ Is $x$ also random, or is it a constant? $\endgroup$ Dec 11, 2020 at 17:20
  • $\begingroup$ It is fixed -- updated my question to make this clear. Thanks! $\endgroup$
    – WazyMaze
    Dec 11, 2020 at 17:29
  • $\begingroup$ It's known that you can generate matrices $O$ by using $O=Q(Q^{T}Q)^{-1/2}$ where the entries of the $n$ by $r$ matrix $Q$ are iid $N(0,1)$. $\endgroup$ Dec 11, 2020 at 18:00
  • $\begingroup$ This is true -- but I could not think of an obvious way to use this fact in a proof. $\endgroup$
    – WazyMaze
    Dec 11, 2020 at 18:09
  • $\begingroup$ Also, I have been able to verify my conjecture experimentally using this construction. $\endgroup$
    – WazyMaze
    Dec 11, 2020 at 18:31

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