Covariance matrix

As a covariance matrix (rarely also: variance -covariance matrix ) is the matrix of all pairwise covariances of the elements of a random vector is called in probability theory. In this respect, this concept generalizes the influence of the variance of a one-dimensional random variable on a multi-dimensional random variable.

, The covariance matrix contains information about the scattering of a random vector and correlation between the components thereof. If a random vector, the associated covariance matrix is given by

It denotes the covariance of the real random variables and. Occasionally, the covariance matrix is ​​denoted by and by.

Is the expected value of the random vector, the following applies

This expectation values ​​of vectors and matrices are to be understood componentwise.

Properties

• The covariance matrix on the main diagonal contains the variances of the individual components of the random vector. Thus all the elements on the main diagonal are non- negative.
• A real covariance matrix is symmetric, since the covariance of two random variables is symmetric.
• The covariance matrix is positive semidefinite: Because of the symmetry of each covariance matrix is diagonalizable by means of principal axis transformation, the diagonal matrix is a covariance matrix again. There are only the variances on the diagonal, the diagonal matrix is thus a positive semi-definite, and thus the original covariance matrix.
• Conversely, each symmetric positive semi-definite matrix, to be construed as a covariance matrix -dimensional random vector.
• Due to the diagonalizability, the eigenvalues ​​( on the diagonal ) are non- negative due to the positive semidefiniteness, covariance matrices can be represented as ellipsoids.
• For all matrices.
• For all vectors.
• Are and uncorrelated random vectors, then applies.
• Are the random variables standardized, so the covariance matrix just contains the correlation coefficients.

To simulate a random vector to obey a given covariance matrix and have the expected value, one determines the decomposition of the covariance matrix (eg the Cholesky decomposition ) and calculates the random vector, where a random vector with independent standard normal components.