Covariance Matrix

The covariance matrix for a random vector $X=(X_1,\dots ,X_n)$ is the $n\times n$ matrix $\Sigma$ defined as:

$${\Sigma }_{ij}=\mathrm{Cov}[X_i,X_j]$$

The covariance matrix captures the variance of each random variable and the covariance between different random variables. The diagonal entries of the covariance matrix represent the variance of each random variable, while the off-diagonal entries represent the covariance between pairs of random variables.