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(This is related to this question)

$Q \in \mathbb{R}^{n\times k}$ is a random matrix where $k$Q$ are orthogonal (i.e. $Q^T Q = I$). To examine $E(QQ^T)$, I conducted monte carlo simulations (using matlab):

[Q R] = qr(randn(n,k),0); 

In other words, I just sampled a $\mathbb{R}^{n\times k}$ matrix from a standard gaussian, then did QR decomposition on it and assumed $Q$ is uniformly distributed in the space where $Q^TQ=I$. Joriki's answer and my simulations aligned so I assume there's nothing majorly wrong with how I obtained samples.

I have two questions (in order of importance)

  1. How does one prove that the $Q$ sampled as above is uniformly distributed in the space where $Q^TQ=I$?
  2. Are there more efficient methods of sampling orthogonal $Q$?
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    That works. When you do gram schmidt orthogonalization on the guassian you get uniform on orthogonal. This corresponds pretty directly to the first column being uniform on $S^n$. You just normalize the gaussian first column and it it uniform on the sphere, then the second should be uniform on vectors orthogonal to first, and conditonally on the first it has a normal distribution that makes the property pretty apparent.2012-04-29
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    @mike I can see that it works and (now that I think of it) that the first column would be uniform on $S^n$. I am still unclear on how the rest of what you state is "pretty apparent". I appreciate the response but can you lay out a fleshed out proof without stating that it's apparent? (I tried doing a proof using gram-schmidt, but couldn't prove that the $k$-th orthonormal vector obtained is uniformly distributed assuming that the $(k-1)$-th vector is uniformly distributed in the orthogonal space.)2012-04-29
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    @mike I think it might be easier if you replied in an answer.2012-04-30

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