Part of a series on Statistics
Correlation and covariance
For random vectors Autocorrelation matrix Cross-correlation matrix Auto-covariance matrix Cross-covariance matrix
For stochastic processes Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
For deterministic signals Autocorrelation function Cross-correlation function Autocovariance function Cross-covariance function
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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 ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T))}$ and ${\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T))}$, each containing random elements whose expected value and variance exist, the cross-correlation matrix of ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ is defined by^[1]: p.337

and has dimensions ${\displaystyle m\times n}$. Written component-wise:

${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }={\begin{bmatrix}\operatorname {E} [X_{1}Y_{1}]&\operatorname {E} [X_{1}Y_{2}]&\cdots &\operatorname {E} [X_{1}Y_{n}]\\\\\operatorname {E} [X_{2}Y_{1}]&\operatorname {E} [X_{2}Y_{2}]&\cdots &\operatorname {E} [X_{2}Y_{n}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\operatorname {E} [X_{m}Y_{1}]&\operatorname {E} [X_{m}Y_{2}]&\cdots &\operatorname {E} [X_{m}Y_{n}]\\\\\end{bmatrix))}$

The random vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ need not have the same dimension, and either might be a scalar value.

Example

For example, if ${\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T))}$ and ${\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)^{\rm {T))}$ are random vectors, then ${\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} ))$ is a ${\displaystyle 3\times 2}$ matrix whose ${\displaystyle (i,j)}$-th entry is ${\displaystyle \operatorname {E} [X_{i}Y_{j}]}$.

Complex random vectors

If ${\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{m})^{\rm {T))}$ and ${\displaystyle \mathbf {W} =(W_{1},\ldots ,W_{n})^{\rm {T))}$ are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of ${\displaystyle \mathbf {Z} }$ and ${\displaystyle \mathbf {W} }$ is defined by

${\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H))]}$

where ${\displaystyle {}^{\rm {H))}$ denotes Hermitian transposition.

Uncorrelatedness

Two random vectors ${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T))}$ and ${\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T))}$ are called uncorrelated if

${\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T))]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T)).}$

They are uncorrelated if and only if their cross-covariance matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} ))$ matrix is zero.

In the case of two complex random vectors ${\displaystyle \mathbf {Z} }$ and ${\displaystyle \mathbf {W} }$ they are called uncorrelated if

${\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H))]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H))}$

and

${\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {T))]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {T)).}$

Properties

Relation to the cross-covariance matrix

The cross-correlation is related to the cross-covariance matrix as follows:

${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\rm {T))]=\operatorname {R} _{\mathbf {X} \mathbf {Y} }-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T))}$

Respectively for complex random vectors:

${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])(\mathbf {W} -\operatorname {E} [\mathbf {W} ])^{\rm {H))]=\operatorname {R} _{\mathbf {Z} \mathbf {W} }-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H))}$