Editing Multivariate random variable (section)

==Correlation and cross-correlation==
===Definitions===
The '''[[Autocorrelation matrix|correlation matrix]]''' (also called '''second moment''') of an <math>n \times 1</math> random vector is an <math>n \times n</math> matrix whose (''i,j'')<sup>th</sup> element is the correlation between the ''i''<sup> th</sup> and the ''j''<sup> th</sup> random variables. The correlation matrix is the expected value, element by element, of the <math>n \times n</math> matrix computed as <math>\mathbf{X} \mathbf{X}^T</math>, where the superscript T refers to the transpose of the indicated vector:<ref name=Papoulis>{{cite book |last=Papoulis |first=Athanasius |title=Probability, Random Variables and Stochastic Processes |publisher=McGraw-Hill |edition=Third |year=1991 |isbn=0-07-048477-5 }}</ref>{{rp|p.190}}<ref name=Gubner/>{{rp|p.334}}

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By extension, the '''cross-correlation 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

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|equation = {{NumBlk||<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[\mathbf{X} \mathbf{Y}^T]</math>|{{EquationRef|Eq.6}}}}
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===Properties===
The correlation matrix is related to the covariance matrix by
:<math>\operatorname{R}_{\mathbf{X}\mathbf{X}} = \operatorname{K}_{\mathbf{X}\mathbf{X}} + \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{X}]^T</math>.
Similarly for the cross-correlation matrix and the cross-covariance matrix:
:<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}} = \operatorname{K}_{\mathbf{X}\mathbf{Y}} + \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^T</math>