Covariance matrix

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> giving the <a href="/facts/Covariance/hILocGNH">covariance</a> between each pair of elements of a given <a href="/facts/Random_vector/qMfooyVf">random vector</a>.
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.
Any <a href="/facts/Covariance/hILocGNH">covariance</a> matrix is <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a> and <a href="/facts/Positive_semi-definite_matrix/Hbr6AuPS">positive semi-definite</a> and its main diagonal contains <a href="/facts/Variance/ULBJKXD1">variances</a> (i.e., the covariance of each element with itself).
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} }}
 
, 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
 or 
 
 
 
 S
 
 
 {\displaystyle S}
 
.

Covariance matrix open-in-new

Covariance matrix