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In probability theory and statistics, a covariance matrix is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances.
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 x {\displaystyle x} and y {\displaystyle y} directions contain all of the necessary information; a 2 × 2 {\displaystyle 2\times 2} matrix would be necessary to fully characterize the two-dimensional variation.
The covariance matrix of a random vector X {\displaystyle \mathbf {X} } is typically denoted by K X X {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }} or Σ {\displaystyle \Sigma } .