Integration on the space of symmetric matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:53:07Z http://mathoverflow.net/feeds/question/95252 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95252/integration-on-the-space-of-symmetric-matrices Integration on the space of symmetric matrices Nimza 2012-04-26T12:43:07Z 2012-09-10T08:58:14Z <p>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 $A \in G$.</p> <p>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 $a_1,\ldots,a_n$, for example the set of all positively defined matrices.</p> <p>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?</p> http://mathoverflow.net/questions/95252/integration-on-the-space-of-symmetric-matrices/95254#95254 Answer by Alex Eskin for Integration on the space of symmetric matrices Alex Eskin 2012-04-26T13:10:00Z 2012-04-26T13:10:00Z <p>I think you are looking for the Haar measure of $SL(n,\mathbb{R})$ in $K A K$ coordinates. You can find it for example in Knapp's book "Representation Theory of Semisimple Groups: An Overview Based on Examples".</p> <p>The short answer: the measure you want is the Haar measure on $O(n) \times O(n)$ times Haar measure on the space of diagonal matrices (viewed as a group under multiplication) times the product of the hyperbolic sines of the positive roots. </p> http://mathoverflow.net/questions/95252/integration-on-the-space-of-symmetric-matrices/95256#95256 Answer by Liviu Nicolaescu for Integration on the space of symmetric matrices Liviu Nicolaescu 2012-04-26T13:34:44Z 2012-09-10T08:58:14Z <p>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</p> <p>$$(2\pi)^{-n(n+1)/4} \int_{G_n} h(A) e^{-({\rm tr} A^2)/4} dA$$</p> <p>$$=\frac{1}{Z_n}\int_{\mathbb{R}^n} h(\lambda_1,\dotsc, \lambda_n) e^{-\frac{1}{4} \sum_{j=1}^n \lambda_j^2} {\prod} |\lambda_j-\lambda_k| d\lambda_1\cdots d\lambda_n$$</p> <p>where the product is taken over $j&lt; k$ and </p> <p>$$Z_n =2^{\frac{n}{2}}n! \prod_{j=1}^n \Gamma(\frac{j}{2}).$$</p> <p>(Sorry for the typesetting clumsiness. There seems to be problem with MathJax.) </p> <p>Now use this formula for </p> <p>$$h(A)= f(g(A)) e^{\frac{{\rm tr} A^2}{4}}.$$</p> <p>For more details see Appendix B of <a href="http://www.nd.edu/~lnicolae/RandCrVal.pdf" rel="nofollow">this paper</a> and the references therein.</p>