This is documentation for Mathematica 9, which was
based on an earlier version of the Wolfram Language.
BUILT-IN MATHEMATICA SYMBOL

# Covariance

Covariance[v1, v2]
gives the covariance between the vectors and .

Covariance[m]
gives the covariance matrix for the matrix m.

Covariance[m1, m2]
gives the covariance matrix for the matrices and .

Covariance[dist]
gives the covariance matrix for the multivariate symbolic distribution dist.

Covariance[dist, i, j]
gives the covariance for the multivariate symbolic distribution dist.

## DetailsDetails

• Covariance[v1, v2] gives the unbiased estimate of the covariance between and .
• The lists and must be the same length.
• Covariance[v1, v2] is equivalent to (v1-Mean[v1]). Conjugate[v2-Mean[v2]]/(Length[v1]-1).
• For a matrix m with columns, Covariance[m] is a × matrix of the covariances between columns of m.
• For an × matrix and an × matrix , Covariance[m1, m2] is a × matrix of the covariances between columns of and columns of .
• Covariance works with SparseArray objects.
• Covariance[dist, i, j] gives Expectation[(xi-i)(xj-j), {x1, x2, ...}dist], where is the i component of the mean of dist.
• Covariance[dist] gives a covariance matrix with the entry given by Covariance[dist, i, j].

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Covariance between two vectors:

 Out[1]=

Covariance matrix for a matrix:

 Out[1]//MatrixForm=

Covariance matrix for two matrices:

 Out[1]//MatrixForm=