markdown 语法支持不是很好

(1) 标准化原始数据
$$
x' = \frac{x-μ}{σ}
$$
第一列

均值 $μ_1 = 0$ , 方差 ${σ_1}^2 = [(1-0)^2 + (2-0)^2 + (-3-0)^2]/3 = 14/3$

第二列

均值 $μ_2 =1/3 $ , 方差 ${σ_2}^2 = [(3-1/3)^2 + (5-1/3)^2 + (-7-1/3)^2]/3 = 248/9$

第三列

均值 $μ_3 =-19/3 $ , 方差 ${σ_3}^2 = [(-7+19/3)^2 + (-14+19/3)^2 + (2+19/3)^2]/3 = 386/9$

则，
$$
\mathbf{X'} = \begin{vmatrix} 0.46291005&0.50800051&-0.10179732\\0.9258201&0.88900089&-1.17066918\\-1.38873015&-1.3970014&1.2724665\\\end{vmatrix}
$$

(2)协方差矩阵
$$
\mathbf{cov(X_{,i}, X_{,j})} = \frac{\sum_{k=1}^m(x_{k,i} - \bar{X_{,i}})(x_{k,j} - \bar{X_{,j}})}{m-1}
$$

$$
\mathbf{X'}.mean(asix=0) = [0,0, -7.401486830834377e-17]
$$

$$
\mathbf{cov(X_{,i}, X_{,j})} = \frac{(\mathbf{X'[:,i-1]} - \mathbf{X'[:,i-1]}.mean()).transpose().dot(\mathbf{X'[:,j-1]} - \mathbf{X'[:,j-1]}.mean())} {m-1}
$$

协方差矩阵(对角线上是各维特征的方差)：
$$
\mathbf{COV} = \begin{vmatrix} \mathbf{cov(X_{,1}, X_{,1})} & \mathbf{cov(X_{,1}, X_{,2})} & \mathbf{cov(X_{,1}, X_{,3})} \\ \mathbf{cov(X_{,2}, X_{,1})} & \mathbf{cov(X_{,2}, X_{,2})} & \mathbf{cov(X_{,2}, X_{,3})} \\ \mathbf{cov(X_{,3}, X_{,1})} &\mathbf{cov(X_{,3}, X_{,2})} &\mathbf{cov(X_{,3}, X_{,3})}\\\end{vmatrix} = \begin{vmatrix} 1.5 & 1.4991357 & -1.44903232 \\ 1.4991357 & 1.5 & -1.43503825 \\ -1.44903232 & -1.43503825 & 1.5 \\\end{vmatrix}
$$