Cross-covariance matrix

Template:Correlation and covariance Template:Confuse Template:Short description 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 $𝐗$ and $𝐘$ is typically denoted by ${\mathrm{K}}_{𝐗𝐘}$ or ${\mathrm{\Sigma }}_{𝐗𝐘}$.

Definition

For random vectors $𝐗$ and $𝐘$, each containing random elements whose expected value and variance exist, the cross-covariance matrix of $𝐗$ and $𝐘$ is defined byTemplate:Rp

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

The cross-covariance matrix is the matrix whose $(i,j)$ entry is the covariance

${\mathrm{K}}_{X_i{Y}_j}=\mathrm{cov}[X_i,Y_j]=\mathrm{E}[(X_i-\mathrm{E}[X_i])(Y_j-\mathrm{E}[Y_j])]$

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

${\mathrm{K}}_{𝐗𝐘}=\left[\begin{array}{cccc}\mathrm{E}[(X_1-\mathrm{E}[X_1])(Y_1-\mathrm{E}[Y_1])]& \mathrm{E}[(X_1-\mathrm{E}[X_1])(Y_2-\mathrm{E}[Y_2])]& \cdots & \mathrm{E}[(X_1-\mathrm{E}[X_1])(Y_n-\mathrm{E}[Y_n])]\\ \\ \mathrm{E}[(X_2-\mathrm{E}[X_2])(Y_1-\mathrm{E}[Y_1])]& \mathrm{E}[(X_2-\mathrm{E}[X_2])(Y_2-\mathrm{E}[Y_2])]& \cdots & \mathrm{E}[(X_2-\mathrm{E}[X_2])(Y_n-\mathrm{E}[Y_n])]\\ \\ ⋮& ⋮& \ddots & ⋮\\ \\ \mathrm{E}[(X_m-\mathrm{E}[X_m])(Y_1-\mathrm{E}[Y_1])]& \mathrm{E}[(X_m-\mathrm{E}[X_m])(Y_2-\mathrm{E}[Y_2])]& \cdots & \mathrm{E}[(X_m-\mathrm{E}[X_m])(Y_n-\mathrm{E}[Y_n])]\end{array}\right]$

Example

For example, if $𝐗={(X_1,X_2,X_3)}^{\mathrm{T}}$ and $𝐘={(Y_1,Y_2)}^{\mathrm{T}}$ are random vectors, then $\mathrm{cov}(𝐗,𝐘)$ is a $3\times 2$ matrix whose $(i,j)$-th entry is $\mathrm{cov}(X_i,Y_j)$.

Properties

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

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

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

Definition for complex random vectors

Template:Main If $𝐙$ and $𝐖$ are complex random vectors, the definition of the cross-covariance matrix is slightly changed. Transposition is replaced by Hermitian transposition:

${\mathrm{K}}_{𝐙𝐖}=\mathrm{cov}(𝐙,𝐖)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\mathrm{E}[(𝐙-{\mathit{\mu }}_{𝐙})(𝐖-{\mathit{\mu }}_{𝐖}{)}^{\mathrm{H}}]$

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

${\mathrm{J}}_{𝐙𝐖}=\mathrm{cov}(𝐙,\overline{𝐖})\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\mathrm{E}[(𝐙-{\mathit{\mu }}_{𝐙})(𝐖-{\mathit{\mu }}_{𝐖}{)}^{\mathrm{T}}]$

Uncorrelatedness

Template:Main Two random vectors $𝐗$ and $𝐘$ are called uncorrelated if their cross-covariance matrix ${\mathrm{K}}_{𝐗𝐘}$ matrix is a zero matrix.Template:Rp

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