computing the Haar measure for O(n) and U(n) groups

Question

My question is about how to compute the Haar measure for O(n) and U(n) groups.

For example, for the conventional parametrization of SO(3) with 3 angels, the Haar measure is

$ dO= cos(\theta_{13})*d\theta_{12}*d\theta_{13}*d\theta_{23}$ .

How one derives this formula? I want to compute the Haar measure for both O(n) and U(n) for general n.

I am a student of physics and I am not quite familiar with Haar measure.

See here for a general description of Haar measure on $O(n)$, and here for the same thing on $U(n)$. — Jim Belk, Jul 18 '15 at 04:24

score 1 · Answer 1 · answered Sep 15 '14 at 23:54

1

You might like these lecture notes. In particular, the second page discusses such calculations.

answered Sep 15 '14 at 23:54

Robin Goodfellow

thank you for the lecture note, though it doesn't contain what I was looking for explicitly, but it helped a bit. I found nice example in the book "Integral Geometry and Geometric Probability" by L.A. Santalo – SAS Sep 22 '14 at 18:49

1 Answers1