 Solution

The off-diagonal auxiliary integrations impose $M^i{}_j=0$ for $i\ne j$, by the Fourier representation of a Dirac delta function. After this constraint, $M=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_N)$ andThe Grassmann Gaussian integral over each off-diagonal pair gives its coefficient. Consequently the ghost determinant isThe prime omits the diagonal directions, which were excluded from the outset. The constant sign depends on the Berezin integral ordering and can be absorbed in normalization. Thus the Vandermonde determinant squared is the eigenvalue measure factor in this matrix diagonalization ghost determinant.

Up to an eigenvalue-independent constant, the remaining integral is $\int {\prod }_{i}d{\lambda }_i{e}^{-S_{\mathrm{eig}}}$, whereThe determinant supplies logarithmic eigenvalue repulsion. An ordering restriction on the eigenvalues changes only a constant factorial; no remaining eigenvalue integral needs to be performed. The off-diagonal bosonic contours and the displayed normalization are understood in the usual Fourier-delta prescription.