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.

2)The same for

$$
\int d Z d Z^\dagger e^{-Tr Z Z^\dagger} \frac{x_1\det e^Z -x_2 \det e^{AZ^\dagger}}{\det(1-x_1e^Zg)\det(1-x_2e^{AZ^\dagger}g)}
$$

where g - arbitrary $GL(N)$ matrix.

This post imported from StackExchange MathOverflow at 2014-07-29 11:48 (UCT), posted by SE-user Sasha