Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Covariance ▷ Cross-covariance matrix

Definition: Let $X = [X_1, \ldots, X_n]^\mathrm{T}$ and $Y = [Y_1, \ldots, Y_m]^\mathrm{T}$ be two random vectors that can or cannot be of equal size. Then, the cross-covariance matrix of $X$ and $Y$ is defined as the $n \times m$ matrix in which the entry $(i,j)$ is the covariance of $X_i$ and $Y_j$:

\[\label{eq:covmat-cross} \Sigma_{XY} = \begin{bmatrix} \mathrm{Cov}(X_1,Y_1) & \ldots & \mathrm{Cov}(X_1,Y_m) \\ \vdots & \ddots & \vdots \\ \mathrm{Cov}(X_n,Y_1) & \ldots & \mathrm{Cov}(X_n,Y_m) \end{bmatrix} = \begin{bmatrix} \mathrm{E}\left[ (X_1-\mathrm{E}[X_1]) (Y_1-\mathrm{E}[Y_1]) \right] & \ldots & \mathrm{E}\left[ (X_1-\mathrm{E}[X_1]) (Y_m-\mathrm{E}[Y_m]) \right] \\ \vdots & \ddots & \vdots \\ \mathrm{E}\left[ (X_n-\mathrm{E}[X_n]) (Y_1-\mathrm{E}[Y_1]) \right] & \ldots & \mathrm{E}\left[ (X_n-\mathrm{E}[X_n]) (Y_m-\mathrm{E}[Y_m]) \right] \end{bmatrix} \; .\]
 
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Metadata: ID: D176 | shortcut: covmat-cross | author: JoramSoch | date: 2022-09-26, 09:45.