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

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

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

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

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

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

My question is how to calculate
\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 \mathrm{K} is a full rank matrix?

Does anyone has suggestion?

Thanks for your help!