I) We will assume that the real n×n matrix A is not necessary symmetric. Therefore the Bosonic Gaussian integral in OP's last formula should read
∫Rndnx exp[−12xiAij xj] = ∫Rndnx exp[−14xi(A+AT)ij xj] = √(2π)ndet(A+AT2),
where we assume that the symmetric part of the matrix
A is positive definite. Hence OP's last formula is basically an integral representation for
det(A)√det(A+AT2).
II) OP may also be interested in the Grassmann integral representation of the Pfaffian
pf(A) ∝ ∫dnθ exp[12θiAij θj] = ∫dnθ exp[14θi(A−AT)ij θj],
where we leave it as an exercise to the reader to fix the correct normalization factor in eq. (3). In the second equality we have used that the Grassmann-numbers anticommute
θiθj = −θjθi.
If we use the integral representation (3) as a definition, then the Pfaffian pf(A) only depends on the antisymmetric part of the n×n matrix A. Let us therefore assume that the matrix A is antisymmetric from now on. One may then prove that the square of the Pfaffian is the determinant:
pf(A)2 = det(A).
This post imported from StackExchange Mathematics at 2015-02-07 08:27 (UTC), posted by SE-user Qmechanic