Within-group and between-group scatter decomposition ($T=W+B$)

For groups with sizes $n_r$, means ${\bar{x}}_r$, and overall mean $\bar{x}$, define $W={\sum }_{r,i}(x_{ri}-{\bar{x}}_r)(x_{ri}-{\bar{x}}_r{)}^T$ and $B={\sum }_r{n}_r({\bar{x}}_r-\bar{x})({\bar{x}}_r-\bar{x}{)}^T$. Expanding $x_{ri}-\bar{x}=(x_{ri}-{\bar{x}}_r)+({\bar{x}}_r-\bar{x})$ gives total scatter $T=W+B$ because each within-group residual sum vanishes. Both matrices are positive semidefinite, and $\mathrm{rank}B\le \min (p,g-1)$. They separate within-group noise from between-group mean separation.