Let $u_i=\sum_j^n M_{ij}c_j$ where $c_1,c_2,\dots,c_n\in\mathbb{R}$ and $M_{ij}\in \mathbb{R}$.
How will this change of variables affect an integral over all possible $M_{ij}$?
$$ I=\int_{-\infty}^\infty \left(\prod_{ij}\mathrm{d}M_{ij}\right) f(\mathbf{M}) $$
More precisely, how will $\mathrm{d}M_{ij}$ change under this change of variables? There are $N^2$ integrals initially, how can I conserve this degree of freedom?