There are a lot of places explaining Haar Distribution but they require some basics of Borel fields, Lie fields, etc. I am not a math major, so I do not have exposure to these concepts.

Here is my understanding of Haar distribution:

Take a $N×N$ matrix, say $M$, of i.i.d. standard Gaussian random variables.One can take a QR decomposition of $M$ and get an orthogonal Matrix $Q$. People claim that the matrix $Q$ is a Haar measure over O(N).

A key property: $Q$is left-invariant, meaning, for any $R∈O(N)$, $Q$and $RQ$ have the same distribution.

What is the proof of this invariance property? ( in terms of matrix algebra, if possible)

  • $\begingroup$ It comes down to the rotational symmetry of the standard Gaussian distribution. Each column of $M$ is a Gaussian vector, and if you premultiply $M$ by a non random orthogonal matrix $R$, the distribution of the columns is not changed. But the columns are independent, so the joint distribution of the columns is not changed, either. That is, the distribution of $RM$ is the same as that of $M$. $\endgroup$ Dec 25, 2017 at 12:51
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    $\begingroup$ You may find this interesting $\endgroup$ Dec 18, 2020 at 14:11


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