Scatter matrix

In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix, for instance of the multivariate normal distribution.
Definition.
Given "n" samples of "m"-dimensional data, represented as the m-by-n matrix, formula_1, the sample mean is
where formula_3 is the "j"-th column of formula_4.
The scatter matrix is the "m"-by-"m" positive semi-definite matrix
where formula_6 denotes matrix transpose, and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as
where formula_8 is the "n"-by-"n" centering matrix.
Application.
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 formula_4 are independently sampled from a multivariate normal distribution, then formula_11 has a Wishart distribution.