Editing Covariance (section)

=== Cross-covariance matrix of real random vectors ===
{{main|Cross-covariance matrix}}

For real [[random vector]]s <math>\mathbf{X} \in \mathbb{R}^m</math> and <math>\mathbf{Y} \in \mathbb{R}^n</math>, the <math>m \times n</math> cross-covariance matrix is equal to<ref name=Gubner/>{{rp|p=336}}

{{Equation box 1
|indent = :
|title =
|equation = {{NumBlk||<math>\begin{align}
  \operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{cov}(\mathbf{X},\mathbf{Y}) 
    &= \operatorname{E}\left[
         (\mathbf{X} - \operatorname{E}[\mathbf{X}])
         (\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^\mathrm{T}
       \right] \\
    &= \operatorname{E}\left[\mathbf{X} \mathbf{Y}^\mathrm{T}\right] - \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^\mathrm{T}
\end{align}</math>|{{EquationRef|Eq.2}}}}
|cellpadding = 6
|border
|border colour = #0073CF
|background colour = #F5FFFA}}

where <math>\mathbf{Y}^{\mathrm T}</math> is the [[transpose]] of the vector (or matrix) <math>\mathbf{Y}</math>.

The <math>(i,j)</math>-th element of this matrix is equal to the covariance <math>\operatorname{cov}(X_i,Y_j)</math> between the {{math|''i''}}-th scalar component of <math>\mathbf{X}</math> and the {{math|''j''}}-th scalar component of <math>\mathbf{Y}</math>. In particular, <math>\operatorname{cov}(\mathbf{Y},\mathbf{X})</math> is the [[transpose]]  of <math>\operatorname{cov}(\mathbf{X},\mathbf{Y})</math>.