Editing Multivariate random variable (section)

===Definitions===
The '''[[covariance matrix]]''' (also called '''second central moment''' or variance-covariance matrix) of an <math>n \times 1</math> random vector is an <math>n \times n</math> [[Matrix (mathematics)|matrix]] whose (''i,j'')<sup>th</sup> element is the [[covariance]] between the ''i''<sup> th</sup> and the ''j''<sup> th</sup> random variables. The covariance matrix is the expected value, element by element, of the <math>n \times n</math> matrix [[matrix multiplication|computed as]] <math>[\mathbf{X}-\operatorname{E}[\mathbf{X}]] [\mathbf{X}-\operatorname{E}[\mathbf{X}]]^T</math>, where the superscript T refers to the transpose of the indicated vector:<ref name=Lapidoth/>{{rp|p. 464}}<ref name=Gubner/>{{rp|p.335}}

{{Equation box 1
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|equation = {{NumBlk||<math>\operatorname{K}_{\mathbf{X}\mathbf{X}} = \operatorname{Var}[\mathbf{X}]=\operatorname{E}[(\mathbf{X}-\operatorname{E}[\mathbf{X}])(\mathbf{X}-\operatorname{E}[\mathbf{X}])^{T}] = \operatorname{E}[\mathbf{X} \mathbf{X}^T] - \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{X}]^T</math>|{{EquationRef|Eq.3}}}}
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By extension, the '''[[cross-covariance matrix]]''' between two random vectors <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> (<math>\mathbf{X}</math> having <math>n</math> elements and <math>\mathbf{Y}</math> having <math>p</math> elements) is the <math>n \times p</math> matrix<ref name=Gubner/>{{rp|p.336}}

{{Equation box 1
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|equation = {{NumBlk||<math>\operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{Cov}[\mathbf{X},\mathbf{Y}]=\operatorname{E}[(\mathbf{X}-\operatorname{E}[\mathbf{X}])(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])^{T}] = \operatorname{E}[\mathbf{X} \mathbf{Y}^T] - \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^T</math>|{{EquationRef|Eq.4}}}}
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where again the matrix expectation is taken element-by-element in the matrix. Here the (''i,j'')<sup>th</sup> element is the covariance between the ''i''<sup> th</sup> element of <math>\mathbf{X}</math> and the ''j''<sup> th</sup> element of <math>\mathbf{Y}</math>.