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I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, i.e., uniform measure on real $UU^T=U^TU=I$, where $A$ is a diagonal matrix, and let's say for simplicity $C$ is symmetric (and hence might as well be diagonal too, WLOG).

Following a suggestion from a previous question about this, I found this note on arxiv which derives an approximation, assuming the the parameters are rational.

Do you know of any other references for this? I get that there are no nice closed form solutions unlike the unitary case, but something exact?

Thanks.

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martin
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  • Hello, I have the same question. I wonder if you found any additional relevant references for this in the meantime? – a06e Jan 24 '25 at 11:42

1 Answers1

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It seems that the $O(n)$ case does not give a determinantal form. However the fact that the Lie algebra of $O(n)$ is the set of $n×n$ anti-symmetric matrices gives similar integral formulas for anti-symmetric matrices. See (1.3) and (1.4) in this paper.

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  • Hello, I have the same question. I wonder if you found any additional relevant references for this in the meantime? – a06e Jan 24 '25 at 11:42