I've been given the following integral (actually a path integral from quantum field theory).
$$ Z(a;N) = \int d^{2n}M.exp(-\frac{1}{2}tr(M^2)-\frac{a}{M}tr(M^4)) $$ where M is a square matrix of size $N^2$. The measure $d^{2n}M$ represents an integral over the real and imaginary parts of each entry of M.
The question then asks one to show that $Z(a;N)/Z(0;N)$ can be reduced to an integral over the eigenvalues $\{\lambda_i\}$ of $M$. It gives the hint that the measure if invariant under diagonalisation.
My question is, what does integrating with respect to $M$ mean here? Does one integrate with respect to each element in the matrix separately? Thanks in advance.