Euclidean real variance

Let $X \in \text{Random}(\mathbb{R}^{N \times 1})$ be a D5210: Random real column matrix such that
 (i) $$\mathbb{E} |X|^2 < \infty$$
The variance of $X$ is the D4571: Real matrix $$\mathbb{E} \left[ (X - \mathbb{E} X) (X - \mathbb{E} X)^T \right]$$

Let $X \in \text{Random}(\mathbb{R}^N)$ be a D4383: Random euclidean real number such that
 (i) $$\mathbb{E} |X|^2 < \infty$$
The variance of $X$ is the D4571: Real matrix $$\begin{bmatrix} \mathbb{E}[(X_1 - \mathbb{E} X_1) (X_1 - \mathbb{E} X_1)] & \mathbb{E}[(X_1 - \mathbb{E} X_1) (X_2 - \mathbb{E} X_2)] & \cdots & \mathbb{E}[(X_1 - \mathbb{E} X_1) (X_N - \mathbb{E} X_N)] \\ \mathbb{E}[(X_2 - \mathbb{E} X_2) (X_1 - \mathbb{E} X_1)] & \mathbb{E}[(X_2 - \mathbb{E} X_2) (X_2 - \mathbb{E} X_2)] & \vdots & \mathbb{E}[(X_2 - \mathbb{E} X_2) (X_N - \mathbb{E} X_N)] \\ \vdots & \cdots & \ddots & \vdots \\ \mathbb{E}[(X_N - \mathbb{E} X_N) (X_1 - \mathbb{E} X_1)] & \mathbb{E}[(X_N - \mathbb{E} X_N) (X_2 - \mathbb{E} X_2)] & \cdots & \mathbb{E}[(X_N - \mathbb{E} X_N) (X_N - \mathbb{E} X_N)] \end{bmatrix}$$