# Cross-covariance matrix

In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of observed empirical values or a finite or infinite number of potential values. The potential values are specified by a theoretical joint probability distribution. Intuitively, the cross-covariance matrix generalizes the notion of covariance to multiple dimensions.

The cross-covariance matrix of two random vectors ${\displaystyle \mathbf {X} }$ and ${\displaystyle \mathbf {Y} }$ is typically denoted by ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}$ or ${\displaystyle \Sigma _{\mathbf {X} \mathbf {Y} }}$.

## Definition

For random vectors ${\displaystyle \mathbf {X} }$  and ${\displaystyle \mathbf {Y} }$ , each containing random elements whose expected value and variance exist, the cross-covariance matrix of ${\displaystyle \mathbf {X} }$  and ${\displaystyle \mathbf {Y} }$  is defined by[1]:p.336

${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {cov} (\mathbf {X} ,\mathbf {Y} ){\stackrel {\mathrm {def} }{=}}\ \operatorname {E} [(\mathbf {X} -\mathbf {\mu _{X}} )(\mathbf {Y} -\mathbf {\mu _{Y}} )^{\rm {T}}]}$

(Eq.1)

where ${\displaystyle \mathbf {\mu _{X}} =\operatorname {E} [\mathbf {X} ]}$  and ${\displaystyle \mathbf {\mu _{Y}} =\operatorname {E} [\mathbf {Y} ]}$  are vectors containing the expected values of ${\displaystyle \mathbf {X} }$  and ${\displaystyle \mathbf {Y} }$ . The vectors ${\displaystyle \mathbf {X} }$  and ${\displaystyle \mathbf {Y} }$  need not have the same dimension, and either might be a scalar value.

The cross-covariance matrix is the matrix whose ${\displaystyle (i,j)}$  entry is the covariance

${\displaystyle \operatorname {K} _{X_{i}Y_{j}}=\operatorname {cov} [X_{i},Y_{j}]=\operatorname {E} [(X_{i}-\operatorname {E} [X_{i}])(Y_{j}-\operatorname {E} [Y_{j}])]}$

between the i-th element of ${\displaystyle \mathbf {X} }$  and the j-th element of ${\displaystyle \mathbf {Y} }$ . This gives the following component-wise definition of the cross-covariance matrix.

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

## 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 ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )}$  is a ${\displaystyle 3\times 2}$  matrix whose ${\displaystyle (i,j)}$ -th entry is ${\displaystyle \operatorname {cov} (X_{i},Y_{j})}$ .

## Properties

For the cross-covariance matrix, the following basic properties apply:[2]

1. ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]-\mathbf {\mu _{X}} \mathbf {\mu _{Y}} ^{\rm {T}}}$
2. ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=\operatorname {cov} (\mathbf {Y} ,\mathbf {X} )^{\rm {T}}}$
3. ${\displaystyle \operatorname {cov} (\mathbf {X_{1}} +\mathbf {X_{2}} ,\mathbf {Y} )=\operatorname {cov} (\mathbf {X_{1}} ,\mathbf {Y} )+\operatorname {cov} (\mathbf {X_{2}} ,\mathbf {Y} )}$
4. ${\displaystyle \operatorname {cov} (A\mathbf {X} +\mathbf {a} ,B^{\rm {T}}\mathbf {Y} +\mathbf {b} )=A\,\operatorname {cov} (\mathbf {X} ,\mathbf {Y} )\,B}$
5. If ${\displaystyle \mathbf {X} }$  and ${\displaystyle \mathbf {Y} }$  are independent (or somewhat less restrictedly, if every random variable in ${\displaystyle \mathbf {X} }$  is uncorrelated with every random variable in ${\displaystyle \mathbf {Y} }$ ), then ${\displaystyle \operatorname {cov} (\mathbf {X} ,\mathbf {Y} )=0_{p\times q}}$

where ${\displaystyle \mathbf {X} }$ , ${\displaystyle \mathbf {X_{1}} }$  and ${\displaystyle \mathbf {X_{2}} }$  are random ${\displaystyle p\times 1}$  vectors, ${\displaystyle \mathbf {Y} }$  is a random ${\displaystyle q\times 1}$  vector, ${\displaystyle \mathbf {a} }$  is a ${\displaystyle q\times 1}$  vector, ${\displaystyle \mathbf {b} }$  is a ${\displaystyle p\times 1}$  vector, ${\displaystyle A}$  and ${\displaystyle B}$  are ${\displaystyle q\times p}$  matrices of constants, and ${\displaystyle 0_{p\times q}}$  is a ${\displaystyle p\times q}$  matrix of zeroes.

## Definition for complex random vectors

If ${\displaystyle \mathbf {Z} }$  and ${\displaystyle \mathbf {W} }$  are complex random vectors, the definition of the cross-covariance matrix is slightly changed. Transposition is replaced by Hermitian transposition:

${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} (\mathbf {Z} ,\mathbf {W} ){\stackrel {\mathrm {def} }{=}}\ \operatorname {E} [(\mathbf {Z} -\mathbf {\mu _{Z}} )(\mathbf {W} -\mathbf {\mu _{W}} )^{\rm {H}}]}$

For complex random vectors, another matrix called the pseudo-cross-covariance matrix is defined as follows:

${\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} (\mathbf {Z} ,{\overline {\mathbf {W} }}){\stackrel {\mathrm {def} }{=}}\ \operatorname {E} [(\mathbf {Z} -\mathbf {\mu _{Z}} )(\mathbf {W} -\mathbf {\mu _{W}} )^{\rm {T}}]}$

## Uncorrelatedness

Two random vectors ${\displaystyle \mathbf {X} }$  and ${\displaystyle \mathbf {Y} }$  are called uncorrelated if their cross-covariance matrix ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}$  matrix is a zero matrix.[1]:p.337

Complex random vectors ${\displaystyle \mathbf {Z} }$  and ${\displaystyle \mathbf {W} }$  are called uncorrelated if their covariance matrix and pseudo-covariance matrix is zero, i.e. if ${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0}$ .

## References

1. ^ a b Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.
2. ^ Taboga, Marco (2010). "Lectures on probability theory and mathematical statistics".