Cross-covariance matrix

For random vectors \mathbf{X} and \mathbf{Y}, each containing random elements whose expected value and variance exist, the cross-covariance matrix of \mathbf{X} and \mathbf{Y} is defined by {{Equation box 1 where \mathbf{\mu_X} = \operatorname{E}[\mathbf{X}] and \mathbf{\mu_Y} = \operatorname{E}[\mathbf{Y}] are vectors containing the expected values of \mathbf{X} and \mathbf{Y}. The vectors \mathbf{X} and \mathbf{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 :\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 \mathbf{X} and the j-th element of \mathbf{Y}. This gives the following component-wise definition of the cross-covariance matrix. : \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 \mathbf{X} = \left( X_1,X_2,X_3 \right)^{\rm T} and \mathbf{Y} = \left( Y_1,Y_2 \right)^{\rm T} are random vectors, then \operatorname{cov}(\mathbf{X},\mathbf{Y}) is a 3 \times 2 matrix whose (i,j)-th entry is \operatorname{cov}(X_i,Y_j). ==Properties==

For the cross-covariance matrix, the following basic properties apply: • \operatorname{cov}(\mathbf{X},\mathbf{Y}) = \operatorname{E}[\mathbf{X} \mathbf{Y}^{\rm T}] - \mathbf{\mu_X} \mathbf{\mu_Y}^{\rm T} • \operatorname{cov}(\mathbf{X},\mathbf{Y}) = \operatorname{cov}(\mathbf{Y},\mathbf{X})^{\rm T} • \operatorname{cov}(\mathbf{X_1} + \mathbf{X_2},\mathbf{Y}) = \operatorname{cov}(\mathbf{X_1},\mathbf{Y}) + \operatorname{cov}(\mathbf{X_2}, \mathbf{Y}) • \operatorname{cov}(A\mathbf{X}+ \mathbf{a}, B^{\rm T}\mathbf{Y} + \mathbf{b}) = A\, \operatorname{cov}(\mathbf{X}, \mathbf{Y}) \,B • If \mathbf{X} and \mathbf{Y} are independent (or somewhat less restrictedly, if every random variable in \mathbf{X} is uncorrelated with every random variable in \mathbf{Y}), then \operatorname{cov}(\mathbf{X},\mathbf{Y}) = 0_{p\times q} where \mathbf{X}, \mathbf{X_1} and \mathbf{X_2} are random p \times 1 vectors, \mathbf{Y} is a random q \times 1 vector, \mathbf{a} is a q \times 1 vector, \mathbf{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==

If \mathbf{Z} and \mathbf{W} are complex random vectors, the definition of the cross-covariance matrix is slightly changed. Transposition is replaced by Hermitian transposition: :\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: :\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 \mathbf{X} and \mathbf{Y} are called uncorrelated if their cross-covariance matrix \operatorname{K}_{\mathbf{X}\mathbf{Y}} matrix is a zero matrix. Complex random vectors \mathbf{Z} and \mathbf{W} are called uncorrelated if their covariance matrix and pseudo-covariance matrix is zero, i.e. if \operatorname{K}_{\mathbf{Z}\mathbf{W}} = \operatorname{J}_{\mathbf{Z}\mathbf{W}} = 0. ==References==