I am integrating the following Gaussian over all possible matrix elements $J_{ij}$: $$ I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm{d}J_{ij} \right)$$
How can I deal with the $\sum_{ij}J_{ij}J_{ji}$ terms? The fact that I am integrating over matrix elements confuses me. Any help or advice is always appreciated, thanks !
Decompose the sum over $i,j$ as $$-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji}=$$ $$\qquad\qquad=\sum_{i}\left[(c-a)J_{ii}^2+bJ_{ii}\right]+\sum_{i<j}\left[-a(J_{ij}^2+J_{ji}^2)+b(J_{ij}+J_{ji})+2cJ_{ij}J_{ji}\right]$$ $$\qquad\qquad\equiv\sum_{i} A_i+\sum_{i<j} B_{ij}.$$ Then perform the Gaussian integrals separately for each term in the sum (assuming $c<a$ and $c^2/a<a$), $$I=\left(\prod_{i=1}^N\int e^{A_i}dJ_{ii}\right)\left(\prod_{i<j=1}^{N}\int\int e^{B_{ij}}dJ_{ij}dJ_{ji}\right)$$ $$\qquad\qquad=\left(\frac{\sqrt{\pi } e^{\frac{b^2}{4 a-4 c}}}{\sqrt{a-c}}\right)^N\left(\frac{\pi e^{\frac{b^2}{2 a-2 c}}}{\sqrt{a^2-c^2}}\right)^{N(N-1)/2}=\pi^{N^2/2}e^{\frac{N^2b^2}{4a-4c}}(a-c)^{-N/2}(a^2-c^2)^{-N(N-1)/2}.$$
