I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size).
This is equivalent to
$$\int_\phi \det \left( e^{RM}\right) dR$$ which is in turn
$$\int_\phi \det \left( \sum_k^\infty \frac{1}{k!} (RM)^k \right) dR$$
But I am not sure how I can proceed further. I would be happy if I get a bound or an approximation too.
To provide a little bit of context, I have a gaussian distribution over $R_{ij}$:
$$ \frac{1}{Z}e^{-tr(R^TR)+tr(RM)} $$
I just wanted to see if I can integrate this over permutation matrices, in which case the first term drops.
