Covariance matrix

The matrix formed from the pairwise covariances of several random variables; more precisely, for the -dimensional vector the covariance matrix is the square matrix , where is the vector of mean values. The components of the covariance matrix are:

and for they are the same as () (that is, the variances of the lie on the principal diagonal). The covariance matrix is a symmetric positive semi-definite matrix. If the covariance matrix is positive definite, then the distribution of is non-degenerate; otherwise it is degenerate. For the random vector the covariance matrix plays the same role as the variance of a random variable. If the variances of the random variables are all equal to 1, then the covariance matrix of is the same as the correlation matrix.

The sample covariance matrix for the sample , where the , , are independent and identically-distributed random -dimensional vectors, consists of the variance and covariance estimators:

where the vector is the arithmetic mean of the . If the are multivariate normally distributed with covariance matrix , then is the maximum-likelihood estimator of ; in this case the joint distribution of the elements of the matrix is called the Wishart distribution; it is one of the fundamental distributions in multivariate statistical analysis by means of which hypotheses concerning the covariance matrix can be tested.