Numerical integration for functions of symmetric matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:53:05Z http://mathoverflow.net/feeds/question/106773 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106773/numerical-integration-for-functions-of-symmetric-matrices Numerical integration for functions of symmetric matrices Kjetil B Halvorsen 2012-09-10T03:42:47Z 2012-09-10T07:52:16Z <p>This is mostly a reference request. I have integrals of the type $$\int_C f(A) (dA)$$ where $f$ is a real-valued function of a positive-(semi)definite matrix argument, and the integration region $C$ is an interval in the cone of positive-definite matrices, such as $C=[0.I]$, where this cone interval denotes the set of all positive-definite matrices with positive eigenvalues all less than one. Other cone intervals could also occur, but in most cases they can be transformed to this or a similar form. Mostly the unctions $f$ will be symmetric functions in the sense that $f(AB)=f(BA)$, where $A$ and $B$ are positive-definite matrices.There must be some papers about this kind of problem?</p> http://mathoverflow.net/questions/106773/numerical-integration-for-functions-of-symmetric-matrices/106778#106778 Answer by an12 for Numerical integration for functions of symmetric matrices an12 2012-09-10T04:45:22Z 2012-09-10T07:52:16Z <p>I think that you could transform the problem through the use of tools from Chapter 5 in the following book</p> <ul> <li><a href="http://projecteuclid.org/euclid.lnms/1196285102" rel="nofollow">Eaton (2007). Multivariate Statistics: A Vector Space Approach</a></li> </ul> <p>There are explicit examples there for integration over spaces of symmetric positive definite matrices.</p>