I couldn't quite understand how we calculate the Gaussian Matrix Integral
$$\mathcal{Z}=\int dM\ e^{-N\text{tr}\left(\frac{1}{2}M^2+JM\right)},$$
where the integration measure $dM$ over the $N$ by $N$ matrix $M$ is defined as
$$dM=\prod_{i=1}^{N}dM_{ii}\prod_{i<j=1}^{N}dM_{ij}dM_{ij}^*.$$
I can see how diagonlazing the matrix $M$
$$M=U\Lambda U^\dagger,\qquad\Lambda=\text{diag}(\lambda_1,\lambda_2,\cdots,\lambda_N),$$
simplifies the discussion by decoupling the action. My main problem is how the measure $dM$ transforms after this unitary transformation and how do we go on about calculating the partition function?