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.
For two random vectors
= (
1
and
, each containing random elements whose expected value and variance exist, the cross-correlation matrix of
is defined by[1]: p.337
and has dimensions
Written component-wise: The random vectors
need not have the same dimension, and either might be a scalar value.
are random vectors, then
matrix whose
-th entry is
are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of
is defined by where
denotes Hermitian transposition.
Two random vectors
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
they are called uncorrelated if and The cross-correlation is related to the cross-covariance matrix as follows: