Covariance Matrix

Definition

Covariance Matrix

The covariance matrix of a random vector with finite second moments and mean is

Its diagonal entries are the variances of individual coordinates; its off-diagonal entries are their pairwise covariances. The matrix records the joint spread of around .

Variance in a direction

For any , the scalar has variance

Thus is symmetric and positive semidefinite. It is positive definite exactly when every non-zero direction has positive variance.