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. When the two random vectors are the same, the cross-covariance matrix is referred to as covariance matrix. 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 ${\displaystyle \Sigma _{\mathbf {X} \mathbf {Y} }}$.

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]: 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.

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

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 ${\displaystyle \operatorname {cov} (X_{i},Y_{j})}$.

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.

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:

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

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]: 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 ${\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0}$.