Evaluation of Multidimensional Integral

Question

I would like to have a closed form expression for the multidimensional integral \begin{equation} \int \mathbf x^\text{T} \mathbf K \mathbf x \exp \left [ - \mathbf x^\text{T} \mathbf K \mathbf x\right ] d\mathbf x. \end{equation} where $\mathbf x \in \mathbb R^d$, $\mathbf K \in \mathbb S^{d\times d}$ is a positive definite matrix, and the integration is over all of $\mathbb R^d$.

I believe such an expression exists, but I don't know how to derive it. Any help would be appreciated.

Answer 1

Set $x=Wy$, where $W$ is unitary and $W^{*}KW=D=diag(\lambda_{1},\cdots,\lambda_{d})$. \begin{eqnarray*} \int\left\langle x,Kx\right\rangle e^{-\left\langle x,Kx\right\rangle }dx & = & \int\left\langle y,W^{*}KWy\right\rangle e^{-\left\langle y,W^{*}KWy\right\rangle }dy\\ & = & \int\left\langle y,Dy\right\rangle e^{-\left\langle y,Dy\right\rangle }dy \end{eqnarray*}

Answer 2

Following James's hint, I arrived at the following conclusion:

\begin{equation} \int_{\mathbb R^d} \mathbf x^\text{T} \mathbf K \mathbf x \exp[-\mathbf x^\text{T} \mathbf K \mathbf x] d\mathbf x = \frac{d \pi^{d/2}}{2\sqrt{|K|}} . \end{equation}