\begin{definition}[Definition:Cross-Covariance Matrix]
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. 
Category:Definitions/Probability Theory
\end{definition}