Scatter matrix

For the notion in quantum mechanics, see scattering matrix.

Definition

Given n samples of m-dimensional data, represented as the m-by-n matrix, ${\displaystyle X=[{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_n]}$, 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 j-th column of ${\displaystyle X}$.

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

${\displaystyle S=\sum _{j=1}^n({\mathbf{x}}_j-\overline{\mathbf{x}})({\mathbf{x}}_j-\overline{\mathbf{x}}{)}^T=\sum _{j=1}^n({\mathbf{x}}_j-\overline{\mathbf{x}})\otimes ({\mathbf{x}}_j-\overline{\mathbf{x}})=\left(\sum _{j=1}^n{\mathbf{x}}_j{\mathbf{x}}_j^T\right)-n\overline{\mathbf{x}}{\overline{\mathbf{x}}}^T}$

where ${\displaystyle (\cdot {)}^T}$ denotes matrix transpose, and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as

${\displaystyle S=X{C}_n{X}^T}$

where ${\displaystyle C_n}$ is the n-by-n centering matrix.