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Covariance Matrix ○꠹｜Definition｜1st｜20251119205401-00-⌔

Covariance matrix - Wikipedia

Covariance matrix

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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 $x$ and $y$ directions contain all of the necessary information; a $2\times 2$ 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 $\mathbf{X}$ is typically denoted by $K_{\mathbf{XX}}$, $\Sigma$ or $S$.

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