# Expectation of semi-orthogonal random projection

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?

• Is $x$ also random, or is it a constant? Dec 11, 2020 at 17:20
• It is fixed -- updated my question to make this clear. Thanks! Dec 11, 2020 at 17:29
• 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 $N(0,1)$. Dec 11, 2020 at 18:00
• This is true -- but I could not think of an obvious way to use this fact in a proof. Dec 11, 2020 at 18:09
• Also, I have been able to verify my conjecture experimentally using this construction. Dec 11, 2020 at 18:31