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 and, each containing random elements whose expected value and variance exist, the cross-correlation matrix of and is defined by
and has dimensions. Written component-wise:
The random vectors and need not have the same dimension, and either might be a scalar value.

Example

For example, if and are random vectors, then
is a matrix whose -th entry is.

Complex random vectors

If and are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of and is defined by
where denotes Hermitian transposition.

Uncorrelatedness

Two random vectors and are called uncorrelated if
They are uncorrelated if and only if their cross-covariance matrix matrix is zero.
In the case of two complex random vectors and 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: