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Yes, O(n) is the n(n-1)/2 dimensional space of orthogonal n by n matrices. Vol(O(n)) is its volume. The integrand in the answer is simply the Jacobian of the singular value decomposition, {s_ i} is just the ordered set of the singular value and the integration is performed on the subset bounded by 1. I may just have missed a factor of 1/2^n because of the sign ambiguity in the svd singular values |
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Yes, O(n) is the n(n-1)/2 dimensional space of orthogonal n by n matrices. The integrand in the answer is simply the Jacobian of the singular value decomposition, {s_ i} is just the ordered set of the singular value and the integration is performed on the subset bounded by 1. |
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