See the covariance matrix sections on the discussion page for more information.

This covariance matrix is a useful tool in many different areas.

There are several ways to define the two covariance matrices.

Similar ideas can be used when is random with uncertainty in the covariance matrix.

Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix.

Suppose that the covariance matrix of the errors is Σ.

The covariance matrix can be used to compute portfolio variance.

A covariance matrix M can be represented as the product .

Extensions of this result can be made for more than two random variables, using the covariance matrix.

As an alternative, many methods have been suggested to improve the estimation of the covariance matrix.

The covariance matrix of T will be the identity matrix.

In effect, P is a square root of the covariance matrix V.

Information about image orientation can be derived by first using the second order central moments to construct a covariance matrix.

For the variance, let denote the covariance matrix of .

The solution is given by computing the two covariance matrices:

Suppose is a random (column) vector with covariance matrix and mean 0.

The eigenvectors of this covariance matrix are therefore called eigenfaces.

The covariance matrix for the corner position is , i.e.

The chi-squared test indicates the difference between observed and expected covariance matrices.

See estimation of covariance matrices for details on the derivation.

Suppose we wish to make inference about a covariance matrix whose prior has a distribution.

A "cleaned" model representation would be to model the mean and covariance matrix directly.

For example, in the case of a Gaussian distribution, this comprises the mean and the covariance matrix.

This means that all of these sources of errors can be represented by a covariance matrix.

The covariance matrix is allowed to be singular (in which case the corresponding distribution has no density).