Covariance Matrix

Given $n$ sets of variates denoted $\{x_1\}$, ..., $\{x_n\}$ , the first-order covariance matrix is defined by

$$V_{ij} = \mathop{\rm cov}\nolimits (x_i,x_j) \equiv \left\langle{(x_i-\mu_i)(x_j-\mu_j)}\right\rangle{},$$

where $\mu_i$ is the Mean. Higher order matrices are given by

$$V_{ij}^{mn} = \left\langle{(x_i-\mu_i)^m(x_j-\mu_j)^n}\right\rangle{}.$$

An individual matrix element $V_{ij}=\mathop{\rm cov}\nolimits (x_i,x_j)$ is called the Covariance of $x_i$ and $x_j$.

See also Correlation (Statistical), Covariance, Variance