Covariance Matrix

Definition

Suppose $(\Omega , \mathcal{F} , \mathbfsf{P} )$ is a probability space and $x: \Omega \to \R ^n$ is a random vector. The covariance matrix of $x$ is the matrix $A \in \R ^{n \times n}$ defined by

\[ A_{ij} = \cov(x_i, x_j) \quad \text{for all } i, j = 1, \dots , n \]