Matrix integral identity - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:33:10Z http://mathoverflow.net/feeds/question/47671 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47671/matrix-integral-identity Matrix integral identity Sasha 2010-11-29T12:43:14Z 2010-11-29T12:43:14Z <p>1) How to prove that $N\times N$ matrix integral over complex matrices $Z$ $$\int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det e^{AZ^\dagger}}{\det(1-x_1e^Z)\det(1-x_2e^{AZ^\dagger})}$$ does not depend on the external Hermitian matrix $A$? $x_1$ and $x_2$ are numbers. The statement is trivial for $1\times1$ case.</p> <p>2)The same for </p> <p>$$\int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det e^{AZ^\dagger}}{\det(1-x_1e^Zg)\det(1-x_2e^{AZ^\dagger}g)}$$</p> <p>where g - arbitrary $GL(N)$ matrix.</p>