Gaussian Function - Multi-dimensional Gaussian Function

Multi-dimensional Gaussian Function

In an -dimensional space a Gaussian function can be defined as

f(x) = \exp(-x^TAx) \;,

where is a column of coordinates, is a positive-definite matrix, and denotes transposition.

The integral of a Gaussian function over the whole -dimensional space is given as

\int_{\mathbb{R}^n}\exp(-x^TBx)dx = \sqrt{\frac{\pi^n}{\det{B}}} \;.

It can be easily calculated by diagonalizing the matrix and changing the integration variables to the eigenvectors of .

More generally a shifted Gaussian function is defined as

f(x) = \exp(-x^TAx+s^Tx) \;,

where is the shift vector and the matrix can be assumed to be symmetric, . The following integrals with this function can be calculated with the same technique,

\int d^nx e^{-x^TBx+v^Tx} = \sqrt{\frac{\pi^n}{\det{B}}} \exp(\frac{1}{4}v^TB^{-1}v)\equiv \mathcal{M}\;.
\int d^n x e^{- x^T B x + v^T x} \left( a^T x \right) = (a^T u) \cdot \mathcal{M}\;,\; {\rm where}\; u = \frac{1}{2} B^{- 1} v \;.
\int d^n x e^{- x^T B x + v^T x} \left( x^T D x \right) = \left( u^T D u + \frac{1}{2} {\rm tr} (D B^{- 1}) \right) \cdot \mathcal{M}\;.
\begin{align} & \int d^n x e^{- x^T A' x + s'^T x} \left( - \frac{\partial}{\partial x} \Lambda \frac{\partial}{\partial x} \right) e^{- x^T A x + s^T x} = \\ & = \left( 2 {\rm tr} (A' \Lambda A B^{- 1}) + 4 u^T A' \Lambda A u - 2 u^T (A' \Lambda s + A \Lambda s') + s'^T \Lambda s \right) \cdot \mathcal{M}\;, \\ & {\rm where} \; u = \frac{1}{2} B^{- 1} v, v = s + s', B = A + A' \;. \end{align}

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