Covariance matrix - Knowlpedia

For a random vector $X=(X_1,\ldots,X_d)$ with finite second moment, its covariance matrix is

\operatorname{Cov}(X)=\mathbb E[(X-\mathbb EX)(X-\mathbb EX)^T].

Its $(i,j)$ entry is the scalar covariance $\operatorname{Cov}(X_i,X_j)$ . Equivalently, it is $\mathbb E[XX^T]-(\mathbb EX)(\mathbb EX)^T$ .

Positivity and degeneracy

For every $a$ , $a^T\operatorname{Cov}(X)a=\operatorname{Var}(a\cdot X)\ge0$ . It is positive definite precisely when no nonzero linear combination $a\cdot X$ is almost surely constant. The deterministic choice of a probability average on a torus gives the same matrix construction for a periodic vector field.

Velocity fluctuations

Reynolds averaging stress is the covariance matrix of a fluctuating velocity. Its sign in a momentum equation depends on which side contains the tensor divergence.