In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix of the multivariate normal distribution. (The scatter matrix is unrelated to the scattering matrix of quantum mechanics.)
Given n samples of m-dimensional data, represented as the m-by-n matrix, , the sample mean is
where is the jth column of .
The scatter matrix is the m-by-m positive semi-definite matrix
where denotes matrix transpose. The scatter matrix may be expressed more succinctly as
where is the n-by-n centering matrix.
The maximum likelihood estimate, given n samples, for the covariance matrix of a multivariate normal distribution can be expressed as the normalized scatter matrix
When the columns of are independently sampled from a multivariate normal distribution, then has a Wishart distribution.