Dear all,
I know how to derive a multidimentional gaussian integration like this:

$\displaystyle \int_{-\infty}^{\infty} \mathrm{d} \mathbf{x} ~ \mathrm{exp} \left\{\mathbf{x}^T \mathbf{G} \mathbf{x} + \mathbf{H}^T \mathbf{x} \right\}$

where $\displaystyle \mathbf{x}=\left\{x_1, x_2, \cdots, x_n\right\} $ and $\displaystyle \mathrm{G} $ is a full rank matrix, and $\displaystyle \mathbf{H}=\left\{H_1, H_2, \cdots, H_n\right\} $

Just diagnolize $\displaystyle \mathrm{G} $ and make a linear transformation:

$\displaystyle \mathbf{x} = \mathbf{V} \mathbf{y}$

where $\displaystyle \mathbf{V} $ is the eigenvector matrix of $\displaystyle \mathrm{G}$.

My question is how to calculate
$\displaystyle \int_{-\infty}^{\infty} \mathrm{d} \mathbf{x} ~ \left( \mathbf{x}^T \mathbf{K} \mathbf{x} \right) \mathrm{exp} \left\{\mathbf{x}^T \mathbf{G} \mathbf{x} + \mathbf{H}^T \mathbf{x}\right\}$

where $\displaystyle \mathrm{K} $ is a full rank matrix?

Does anyone has suggestion?

Thanks for your help!