Definition:Cross-Covariance Matrix

Definition

Let $\mathbf X = \tuple {X_1, X_2, \ldots, X_n}^T$ and $\mathbf Y = \tuple {Y_1, Y_2, \ldots, Y_m}^T$ be random vectors.

Then the cross-covariance matrix of $\mathbf X$ and $\mathbf Y$ is defined by:

$\cov {\mathbf X, \mathbf Y} = \expect {\paren {\mathbf X - \expect {\mathbf X} } \paren {\mathbf Y - \expect {\mathbf Y} }^T}$

where this expectation exists.