$$
\begin{gathered}
X=\begin{pmatrix}
x_{1} & x_{2} & \cdots & x_{N}
\end{pmatrix}^{T}_{N \times p}=\begin{pmatrix}
x_{1}^{T} \ x_{2}^{T} \ \vdots \ x_{N}^{T}
\end{pmatrix}=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1p} \ x_{21} & x_{22} & \cdots & x_{2p} \ \vdots & \vdots & & \vdots \ x_{N1} & x_{N2} & \cdots & x_{NP}
\end{pmatrix}_{N \times p}\
x_{i}\in \mathbb{R}^{p},i=1,2,\cdots ,N\
记1_{N}=\begin{pmatrix}1 \ 1 \ \vdots \ 1\end{pmatrix}_{N \times 1}
\end{gathered}
$$
对于样本均值
$$
\begin{aligned}
\bar{x}&=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}\
&=\frac{1}{N}\begin{pmatrix}
x_{1} & x_{2} & \cdots & x_{N}
\end{pmatrix}\begin{pmatrix}1 \ 1 \ \vdots \ 1\end{pmatrix}_{N \times 1}\
&=\frac{1}{N}X^{T}1_{N}
\end{aligned}
$$
对于样本方差
$$
\begin{aligned}
S&=\frac{1}{N}\sum\limits_{i=1}^{N}(x_{i}-\bar{x})(x_{i}-\bar{x})^{T}
\end{aligned}
$$
对于$\sum\limits_{i=1}^{N}(x_{i}-\bar{x})$有
$$
\begin{aligned}
\sum\limits_{i=1}^{N}(x_{i}-\bar{x})&=\begin{pmatrix}
x_{1}-\bar{x} & x_{2}-\bar{x} & \cdots & x_{N}-\bar{x}
\end{pmatrix}\
&=\begin{pmatrix}
x_{1} & x_{2} & \cdots & x_{N}
\end{pmatrix}-\begin{pmatrix}
\bar{x} & \bar{x} & \cdots & \bar{x}
\end{pmatrix}\
&=X^{T}-\bar{x}\begin{pmatrix}1 & 1 & \cdots & 1\end{pmatrix}\
&=X^{T}-\bar{x}1_{N}^{T}\
&=X^{T}- \frac{1}{N}X^{T}1_{N}1_{N}^{T}\
&=X^{T}\left(\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\right)\
\end{aligned}
$$
带回原式
$$
\begin{aligned}
S&=\frac{1}{N}\begin{pmatrix}
x_{1}-\bar{x} & x_{2}-\bar{x} & \cdots & x_{N}-\bar{x}
\end{pmatrix}\begin{pmatrix}
(x_{1}-\bar{x})^{T} \ (x_{2}-\bar{x})^{T} \ \vdots \ (x_{N}-\bar{x})^{T}
\end{pmatrix}\
&=\frac{1}{N}X^{T}\left(\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\right)\cdot (\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T})^{T}X\
\end{aligned}
$$
记$\begin{aligned} \mathbb{H}=\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\end{aligned}$（$\mathbb{H}$也被称为中心矩阵），上式为
$$
\begin{aligned}
S&=\frac{1}{N}X^{T}\left(\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\right)\cdot (\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T})^{T}X\
&=\frac{1}{N}X^{T}\mathbb{H}\cdot \mathbb{H}X
\end{aligned}
$$
对于$\mathbb{H}^{T}$有
$$
\begin{aligned}
\mathbb{H}^{T}&=(\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T})^{T}\
&=\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\
&=\mathbb{H}
\end{aligned}
$$
对于$\mathbb{H}^{2}$有
$$
\begin{aligned}
\mathbb{H}^{2}&=\mathbb{H} \cdot \mathbb{H}\
&=\left(\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\right)\left(\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\right)\
&=\mathbb{I}{N}- \frac{2}{N}1{N}1_{N}^{T}+ \frac{1}{N^{2}}1_{N}1_{N}^{T}1_{N}1_{N}^{T}
\end{aligned}
$$
对于$1_{N}1_{N}^{T}$
$$
\begin{aligned}
1_{N}1_{N}^{T}&=\begin{pmatrix}
1 \ \vdots \ 1
\end{pmatrix}\begin{pmatrix}
1 & \cdots & 1
\end{pmatrix}=\begin{pmatrix}
1 & \cdots & 1 \ \vdots & & \vdots \ 1 & \cdots & 1
\end{pmatrix}\
1_{N}1_{N}^{T}1_{N}1_{N}^{T}&=\begin{pmatrix}
1 & \cdots & 1 \ \vdots & & \vdots \ 1 & \cdots & 1
\end{pmatrix}\begin{pmatrix}
1 & \cdots & 1 \ \vdots & & \vdots \ 1 & \cdots & 1
\end{pmatrix}\
&=\begin{pmatrix}
N & \cdots & N \ \vdots & & \vdots \ N & \cdots & N
\end{pmatrix}
\end{aligned}
$$
带回$\mathbb{H}^{2}$有
$$
\begin{aligned}
\mathbb{H}^{2}&=\mathbb{I}{N}- \frac{2}{N}1{N}1_{N}^{T}+ \frac{1}{N^{2}}1_{N}1_{N}^{T}1_{N}1_{N}^{T}\
&=\mathbb{I}_{N}- \frac{2}{N}\begin{pmatrix}
1 & \cdots & 1 \ \vdots & & \vdots \ 1 & \cdots & 1
\end{pmatrix}+ \frac{1}{N^{2}}\begin{pmatrix}
N & \cdots & N \ \vdots & & \vdots \ N & \cdots & N
\end{pmatrix}\
&=\mathbb{I}_{N}- \frac{2}{N}\begin{pmatrix}
1 & \cdots & 1 \ \vdots & & \vdots \ 1 & \cdots & 1
\end{pmatrix}+ \frac{1}{N}\begin{pmatrix}
1 & \cdots & 1 \ \vdots & & \vdots \ 1 & \cdots & 1
\end{pmatrix}\
&=\mathbb{I}_{N}- \frac{1}{N}\begin{pmatrix}
1 & \cdots & 1 \ \vdots & & \vdots \ 1 & \cdots & 1
\end{pmatrix}\
&=\mathbb{I}{N}- \frac{1}{N}1{N}1_{N}^{T}\
&=\mathbb{H}
\end{aligned}
$$
因此有$\mathbb{H}^{n}=\mathbb{H}$，带回$S$
$$
\begin{aligned}
S&=\frac{1}{N}X^{T}\mathbb{H}\cdot \mathbb{H}X\
&=\frac{1}{N}X^{T}\mathbb{H}X
\end{aligned}
$$
这里中心矩阵$\mathbb{H}$的几何意义是，对于一个数据集$X$，$X \mathbb{H}$可以认为是将数据集平移到坐标轴原点，$\mathbb{H}$就是这个起到平移作用的矩阵