= Matrix diagonalization ghost determinant
{title2=$\det{}'\operatorname{ad}_{\operatorname{diag}\lambda}\propto\prod_{i<j}(\lambda_i-\lambda_j)^2$}
Imposing vanishing off-diagonal entries with auxiliary integrations and an off-diagonal <Grassmann Gaussian integral> gives a determinant of the commutator map on the off-diagonal matrix units. Its eigenvalues are $\lambda_i-\lambda_j$, $i\ne j$. Up to a constant sign, their product is the squared <Vandermonde determinant>. It contributes $-2\sum_{i<j}\log|\lambda_i-\lambda_j|$ to the eigenvalue action, encoding repulsion between distinct <eigenvalues>. Diagonal zero modes and constant normalization factors must be treated separately.
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