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<math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}</math> is a <math>3 \times 2</math> matrix whose <math>(i,j)</math>-th entry is <math>\operatorname{E}[X_i Y_j]</math>.
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If <math>\mathbf{Z} = (Z_1,\ldots,Z_m)^{\rm T}</math> and <math>\mathbf{W} = (W_1,\ldots,W_n)^{\rm T}</math> are [[complex random vector|complex random vectors]], each containing random variables whose expected value and variance exist, the cross-correlation matrix of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> is defined by
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:<math>\operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] = \operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^{\rm T}.</math>
They are uncorrelated if and only if their cross-covariance matrix <math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}</math> matrix is zero.
In the case of two [[complex random vector|complex random vectors]] <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> they are called uncorrelated if
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==Properties==
The cross-correlation is related to the ''cross-covariance matrix'' as follows:
:<math>\operatorname{K}_{\mathbf{X}\mathbf{Y}} = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{Y} - \operatorname{E}[\mathbf{Y}])^{\rm T}] = \operatorname{R}_{\mathbf{X}\mathbf{Y}} - \operatorname{E}[\mathbf{X}] \operatorname{E}[\mathbf{Y}]^{\rm T}</math>
: Respectively for complex random vectors:
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