Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 40 / 3 / ii / Solution

Choose the orientation of the Berezin integral so thatHere the product has increasing $i$; this explicitly fixes the otherwise convention-dependent overall sign in the compact measure notation.

Since $A$ is a diagonalizable matrix, write $A=S\Lambda S^{-1}$ with $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_N)$. Make the independent changes $\eta =S\xi$ and $\bar{\eta }=\bar{\xi }{S}^{-1}$. The Grassmann change-of-variables formula gives the two factors $(\det S{)}^{-1}$ and $\det S$, so the complete measure is unchanged. The exponent becomes ${\sum }_i{\lambda }_i{\bar{\xi }}_i{\xi }_i$. Each summand is even and squares to zero, and the different even summands commute. Consequently,Only the term containing every generator survives the Berezin integral. Hence the Grassmann Gaussian integral isZero eigenvalues give zero on both sides, so invertibility of $A$ is unnecessary. In fact the identity extends to all ordinary matrices: the top-degree coefficient of the exponential is the alternating determinant expansion. The assumption that $A$ is a diagonalizable matrix makes the proof especially transparent.