Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi\_{A}(x)$ be its characteristic polynomial:


$$\chi\_{A}(-x) = \det (A+xI) = x^n + a\_{1}x^{n-1}+\ldots+a\_{n-1}x + a\_{n}.$$


Let $g(A) = ( a\_1, \ldots, a\_n )$ for any $A \in G$.


Consider a function $f \colon \mathbb{R}^n \to \mathbb{R}$ and define an integral


$$I =\int\limits\_{H} f(g(A)) \, \mu(dA),$$


where $H$ is some subset of $G$ well characterized by $a\_1,\dots,a\_n$, for example the set of all positively defined matrices.


My question is: how to reduce the integration with respect to $\mu$ to the integration with respect to the Lebesgue measure on the space of eigenvalues? I did't find an easy way. Maybe I have to use some generalized version of coarea formula to split the integration on the integration with respect to the Haar measure on $O(n)$ plus the integration on the space of eigenvalues?



Integration on the space of symmetric matrices
There is a Weyl integration formula that deals with this problem. $\newcommand{\bR}{\mathbb{R}}$ Denote by $G\_n$ the space of symmetric $n\times n$ real matrices. It states that for any $O(n)$-invariant $h:G\_n\to \bR$ we have


$$ (2\pi)^{-\frac{n(n+1)}{4}} \int\_{G\_n} h(A) e^{-\frac{{\rm tr} A^2}{4}} dA $$


$$
=\frac{1}{Z\_n}\int\_{\mathbb{R}^n} h(\lambda\_1,\dotsc, \lambda\_n) e^{-\frac{1}{4} \sum\_{j=1}^n \lambda\_j^2} \; {\prod\_{j<k}} |\lambda\_j-\lambda\_k| \; d\lambda\_1\cdots d\lambda\_n$$


where


$$Z\_n =2^{\frac{n}{2}}n! \prod\_{j=1}^n \Gamma(\frac{j}{2}).$$


Now, use this formula for


$$ h(A)= f(g(A))\, e^{\frac{{\rm tr} A^2}{4}}. $$


For more details see Appendix C of [this paper](http://www.nd.edu/%7Elnicolae/RandCrVal.pdf) and the references therein.


