# Scatter matrix

For the notion in quantum mechanics, see scattering 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, $X=[\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n}]$ , the sample mean is

${\overline {\mathbf {x} }}={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {x} _{j}$ where $\mathbf {x} _{j}$ is the j-th column of $X$ .

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

$S=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})(\mathbf {x} _{j}-{\overline {\mathbf {x} }})^{T}=\sum _{j=1}^{n}(\mathbf {x} _{j}-{\overline {\mathbf {x} }})\otimes (\mathbf {x} _{j}-{\overline {\mathbf {x} }})=\left(\sum _{j=1}^{n}\mathbf {x} _{j}\mathbf {x} _{j}^{T}\right)-n{\overline {\mathbf {x} }}{\overline {\mathbf {x} }}^{T}$ where $T$ denotes matrix transpose, and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as

$S=X\,C_{n}\,X^{T}$ where $\,C_{n}$ 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

$C_{ML}={\frac {1}{n}}S.$ When the columns of $X$ are independently sampled from a multivariate normal distribution, then $S$ has a Wishart distribution.