Let $X, Y \in \text{Random}(\mathbb{R}^{N \times 1})$ each be a D5210: Random real column matrix such that

(i)	\begin{equation} \mathbb{E} |X|^2, \mathbb{E} |Y|^2 < \infty \end{equation}

The covariance of $(X, Y)$ is the D4571: Real matrix \begin{equation} \mathbb{E} \left[ (X - \mathbb{E} X) (Y - \mathbb{E} Y)^T \right] \end{equation}

Let $X, Y \in \text{Random}(\mathbb{R}^N)$ each be a D4383: Random euclidean real number such that

(i)	\begin{equation} \mathbb{E} |X|^2, \mathbb{E} |Y|^2 < \infty \end{equation}

The covariance of $(X, Y)$ is the D4571: Real matrix \begin{equation} \begin{bmatrix} \mathbb{E}[(X_1 - \mathbb{E} X_1) (Y_1 - \mathbb{E} Y_1)] & \mathbb{E}[(X_1 - \mathbb{E} X_1) (Y_2 - \mathbb{E} Y_2)] & \cdots & \mathbb{E}[(X_1 - \mathbb{E} X_1) (Y_N - \mathbb{E} Y_N)] \\ \mathbb{E}[(X_2 - \mathbb{E} X_2) (Y_1 - \mathbb{E} Y_1)] & \mathbb{E}[(X_2 - \mathbb{E} X_2) (Y_2 - \mathbb{E} Y_2)] & \vdots & \mathbb{E}[(X_2 - \mathbb{E} X_2) (Y_N - \mathbb{E} Y_N)] \\ \vdots & \cdots & \ddots & \vdots \\ \mathbb{E}[(X_N - \mathbb{E} X_N) (Y_1 - \mathbb{E} Y_1)] & \mathbb{E}[(X_N - \mathbb{E} X_N) (Y_2 - \mathbb{E} Y_2)] & \cdots & \mathbb{E}[(X_N - \mathbb{E} X_N) (Y_N - \mathbb{E} Y_N)] \end{bmatrix} \end{equation}