Cross-correlation matrix

The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms.
Definition.
For two random vectors formula_1 and formula_2, each containing random elements whose expected value and variance exist, the cross-correlation matrix of formula_3 and formula_4 is defined by
 \triangleq\ \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}]</math>
and has dimensions formula_5. Written component-wise:
The random vectors formula_3 and formula_4 need not have the same dimension, and either might be a scalar value.
Example.
For example, if formula_9 and formula_10 are random vectors, then
formula_11 is a formula_12 matrix whose formula_13-th entry is formula_14.
Cross-correlation matrix of complex random vectors.
If formula_15 and formula_16 are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of formula_17 and formula_18 is defined by
where formula_20 denotes Hermitian transposition.
Uncorrelatedness.
Two random vectors formula_21 and formula_22 are called uncorrelated if
They are uncorrelated if and only if their cross-covariance matrix formula_24 matrix is zero.
In the case of two complex random vectors formula_17 and formula_18 they are called uncorrelated if
and
Properties.
Relation to the cross-covariance matrix.
The cross-correlation is related to the "cross-covariance matrix" as follows: