Editing Normal distribution (section)

===== Vector form =====
A similar formula can be written for the sum of two vector quadratics: If '''x''', '''y''', '''z''' are vectors of length ''k'', and '''A''' and '''B''' are [[symmetric matrix|symmetric]], [[invertible matrices]] of size <math display=inline>k\times k</math>, then

<math display=block>
\begin{align}
& (\mathbf{y}-\mathbf{x})'\mathbf{A}(\mathbf{y}-\mathbf{x}) + (\mathbf{x}-\mathbf{z})' \mathbf{B}(\mathbf{x}-\mathbf{z}) \\
= {} & (\mathbf{x} - \mathbf{c})'(\mathbf{A}+\mathbf{B})(\mathbf{x} - \mathbf{c}) + (\mathbf{y} - \mathbf{z})'(\mathbf{A}^{-1} + \mathbf{B}^{-1})^{-1}(\mathbf{y} - \mathbf{z})
\end{align}
</math>

where

<math display=block>\mathbf{c} = (\mathbf{A} + \mathbf{B})^{-1}(\mathbf{A}\mathbf{y} + \mathbf{B} \mathbf{z})</math>

The form '''x'''′ '''A''' '''x''' is called a [[quadratic form]] and is a [[scalar (mathematics)|scalar]]:
<math display=block>\mathbf{x}'\mathbf{A}\mathbf{x} = \sum_{i,j}a_{ij} x_i x_j</math>
In other words, it sums up all possible combinations of products of pairs of elements from '''x''', with a separate coefficient for each. In addition, since <math display=inline>x_i x_j = x_j x_i</math>, only the sum <math display=inline>a_{ij} + a_{ji}</math> matters for any off-diagonal elements of '''A''', and there is no loss of generality in assuming that '''A''' is [[symmetric matrix|symmetric]]. Furthermore, if '''A''' is symmetric, then the form <math display=inline>\mathbf{x}'\mathbf{A}\mathbf{y} = \mathbf{y}'\mathbf{A}\mathbf{x}.</math>