# Scatter matrix

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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, ${\displaystyle X=[\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n}]}$, the sample mean is

${\displaystyle {\overline {\mathbf {x} }}={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {x} _{j}}$

where ${\displaystyle \mathbf {x} _{j}}$ is the jth column of ${\displaystyle X\,}$.

The scatter matrix is the m-by-m positive semi-definite matrix

${\displaystyle S=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})(\mathbf {x} _{j}-{\overline {\mathbf {x} }})'}$

where ${\displaystyle {\,}'}$ denotes matrix transpose. The scatter matrix may be expressed more succinctly as

${\displaystyle S=X\,C_{n}\,X\,'}$

where ${\displaystyle \,C_{n}}$ 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

${\displaystyle C_{ML}={\frac {1}{n}}S.}$

When the columns of ${\displaystyle X\,}$ are independently sampled from a multivariate normal distribution, then ${\displaystyle S\,}$ has a Wishart distribution.