Covariance matrix

In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector.

Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the ${\displaystyle x}$ and ${\displaystyle y}$ directions contain all of the necessary information; a ${\displaystyle 2\times 2}$ matrix would be necessary to fully characterize the two-dimensional variation.

Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself).

The covariance matrix of a random vector ${\displaystyle \mathbf {X} }$ is typically denoted by ${\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }}$, ${\displaystyle \Sigma }$ or ${\displaystyle S}$.