Editing Covariance matrix (section)

{{Use American English|date = April 2019}}
{{Short description|Measure of covariance of components of a random vector}}
{{Confuse|Cross-covariance matrix}}
[[Image:Gaussian-2d.png|thumb|right|A [[bivariate Gaussian distribution|bivariate Gaussian probability density function]] centered at (0, 0), with covariance matrix given by
<math>\begin{bmatrix}
1 & 0.5\\
0.5 & 1
\end{bmatrix}</math>]]
[[Image:GaussianScatterPCA.svg|thumb|right|Sample points from a [[bivariate Gaussian distribution]] with a standard deviation of 3 in roughly the lower left–upper right direction and of 1 in the orthogonal direction. Because the ''x'' and ''y'' components co-vary, the variances of <math>x</math> and <math>y</math> do not fully describe the distribution. A <math>2 \times 2</math> covariance matrix is needed; the directions of the arrows correspond to the [[eigenvector]]s of this covariance matrix and their lengths to the square roots of the [[eigenvalues]].]]
{{Correlation and covariance}}

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 (mathematics)|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 <math>x</math> and <math>y</math> directions contain all of the necessary information; a <math>2 \times 2</math> matrix would be necessary to fully characterize the two-dimensional variation.

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

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