Editing Gaussian function (section)

== Multi-dimensional Gaussian function ==
{{main|Multivariate normal distribution}}
In an <math>n</math>-dimensional space a Gaussian function can be defined as
<math display="block">f(x) = \exp(-x^\mathsf{T} C x),</math>
where <math>x = \begin{bmatrix} x_1 & \cdots & x_n\end{bmatrix}</math> is a column of <math>n</math> coordinates, <math>C</math> is a [[positive-definite matrix|positive-definite]] <math>n \times n</math> matrix, and <math>{}^\mathsf{T}</math> denotes [[transpose|matrix transposition]].

The integral of this Gaussian function over the whole <math>n</math>-dimensional space is given as
<math display="block">\int_{\R^n} \exp(-x^\mathsf{T} C x) \, dx = \sqrt{\frac{\pi^n}{\det C}}.</math>

It can be easily calculated by diagonalizing the matrix <math>C</math> and changing the integration variables to the eigenvectors of <math>C</math>.

More generally a shifted Gaussian function is defined as
<math display="block">f(x) = \exp(-x^\mathsf{T} C x + s^\mathsf{T} x),</math>
where <math>s = \begin{bmatrix} s_1 & \cdots & s_n\end{bmatrix}</math> is the shift vector and the matrix <math>C</math> can be assumed to be symmetric, <math>C^\mathsf{T} = C</math>, and positive-definite. The following integrals with this function can be calculated with the same technique:
<math display="block">\int_{\R^n} e^{-x^\mathsf{T} C x + v^\mathsf{T}x} \, dx = \sqrt{\frac{\pi^n}{\det{C}}} \exp\left(\frac{1}{4} v^\mathsf{T} C^{-1} v\right) \equiv \mathcal{M}.</math>
<math display="block">\int_{\mathbb{R}^n} e^{- x^\mathsf{T} C x + v^\mathsf{T} x} (a^\mathsf{T} x) \, dx = (a^T u) \cdot \mathcal{M}, \text{ where } u = \frac{1}{2} C^{-1} v.</math>
<math display="block">\int_{\mathbb{R}^n} e^{- x^\mathsf{T} C x + v^\mathsf{T} x} (x^\mathsf{T} D x) \, dx = \left( u^\mathsf{T} D u + \frac{1}{2} \operatorname{tr} (D C^{-1}) \right) \cdot \mathcal{M}.</math>
<math display="block">\begin{align}
& \int_{\mathbb{R}^n} e^{- x^\mathsf{T} C' x + s'^\mathsf{T} x} \left( -\frac{\partial}{\partial x} \Lambda \frac{\partial}{\partial x} \right) e^{-x^\mathsf{T} C x + s^\mathsf{T} x} \, dx \\
& \qquad = \left( 2 \operatorname{tr}(C' \Lambda C B^{- 1}) + 4 u^\mathsf{T} C' \Lambda C u - 2 u^\mathsf{T} (C' \Lambda s + C \Lambda s') + s'^\mathsf{T} \Lambda s \right) \cdot \mathcal{M},
\end{align}</math>
where <math display="inline">u = \frac{1}{2} B^{- 1} v,\ v = s + s',\ B = C + C'.</math>