# Scatter matrix

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.)

## Definition

Given *n* samples of *m*-dimensional data, represented as the *m*-by-*n* matrix, , the sample mean is

where is the *j*th 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.

## 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 are independently sampled from a multivariate normal distribution, then has a Wishart distribution.

## See also