协方差矩阵的定义
设一个随机向量为\(\mathbf{x} \in \mathbb{R}^\mathrm{N}\),其均值为\(\bar{\mathbf{x}}\),则令\(\mathbf{y} = \mathbf{x} - \bar{\mathbf{x}}\),则随机向量\(\mathbf{x}\)的协方差定义为:
\[\Sigma_{\mathbf{x}} = \begin{bmatrix} \sigma(x_1,x_1) & \dotsb & \sigma(x_1,x_N) \\ \vdots & \ddots & \vdots \\ \sigma(x_N,x_1) & \dotsb & \sigma(x_N,x_N) \end{bmatrix} \in \mathbb{R}^{\mathrm{N} \times \mathrm{N}} \]