Gaussian Integrals

The diagonalized case

$$ \begin{eqnarray} Z_0 &=& \int d^n z \exp\left(-\frac{1}{2} z^\mathrm{T} D z\right) \\ &=& \prod_i \int d z_i \exp\left(-\frac{1}{2} \lambda_i z_i^2\right) \\ &=& \prod_i \sqrt{\frac{2\pi}{\lambda_i}} \\ &=& \sqrt{\frac{(2\pi)^n}{\det A}}. \end{eqnarray} $$

For an arbitrary matrix $A$,

$$ Z_J = \int d^n x \exp\left(-\frac{1}{2} x^\mathrm{T} A x + J^\mathrm{T} x\right). $$

$$ \begin{eqnarray} Z_J &=& \int d^n y \exp\left(-\frac{1}{2} {y}^\mathrm{T} A y + \frac{1}{2} J^\mathrm{T}A^{-1}J\right) \\ &=& \sqrt{\frac{(2\pi)^n}{\det A}} \exp\left(\frac{1}{2} J^\mathrm{T}A^{-1}J\right). \end{eqnarray} $$