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 = [ x 1 , x 2 , … , x n ] {\displaystyle X=[\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n}]} , the sample mean is
x ¯ = 1 n ∑ j = 1 n x j {\displaystyle {\overline {\mathbf {x} }}={\frac {1}{n}}\sum _{j=1}^{n}\mathbf {x} _{j}}
where x j {\displaystyle \mathbf {x} _{j}} is the j-th column of X {\displaystyle X} . The scatter matrix is the m-by-m positive semi-definite matrix
S = ∑ j = 1 n ( x j − x ¯ ) ( x j − x ¯ ) T = ∑ j = 1 n ( x j − x ¯ ) ⊗ ( x j − x ¯ ) = ( ∑ j = 1 n x j x j T ) − n x ¯ x ¯ T {\displaystyle 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 {\displaystyle (\cdot )^{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 {\displaystyle S=X\,C_{n}\,X^{T}}
where C n {\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
C M L = 1 n S . {\displaystyle C_{ML}={\frac {1}{n}}S.}
When the columns of X {\displaystyle X} are independently sampled from a multivariate normal distribution, then S {\displaystyle S} has a Wishart distribution.
See also Estimation of covariance matrices Sample covariance matrix Wishart distribution Outer product— X X ⊤ {\displaystyle XX^{\top }} or X⊗X is the outer product of X with itself. Gram matrix
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