Centering Matrix

Definition

$C=I-\frac{1}{n}{J}_n$ where $J_n=1_n{1}^{⊺}$ is $n\times n$ matrix where all elements are $1$

Summing Vector

$1_n=(1,1,\dots ,1{)}^{⊺}$

Facts

$(n-1)s^2={\mathbf{x}}^{⊺}C\mathbf{x}$ where $s^2$ is a sample variance of $\mathbf{x}$