协方差矩阵-（来自维基百科）

假设{\displaystyle X}

是以{\displaystyle
n}

个随机变数组成的列向量，
{\displaystyle \mathbf {X} ={\begin{bmatrix}X_{1}\\X_{2}\\\vdots
\\X_{n}\end{bmatrix}}}


并且{\displaystyle \mu _{i}}

是{\displaystyle
X_{i}}

的期望值，即, {\displaystyle
\mu _{i}=\mathrm {E} (X_{i})}

。协方差矩阵的第{\displaystyle
(i,j)}

项（第{\displaystyle
(i,j)}

项是一个协方差）被定义为如下形式：
{\displaystyle \Sigma _{ij}=\mathrm
{cov} (X_{i},X_{j})=\mathrm {E} {\begin{bmatrix}(X_{i}-\mu _{i})(X_{j}-\mu _{j})\end{bmatrix}}}


而协方差矩阵为：
{\displaystyle \Sigma =\mathrm {E}
\left[\left(\mathbf {X} -\mathrm {E} [\mathbf {X} ]\right)\left(\mathbf {X} -\mathrm {E} [\mathbf {X} ]\right)^{\rm {T}}\right]}


{\displaystyle ={\begin{bmatrix}\mathrm {E} [(X_{1}-\mu _{1})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{1}-\mu
_{1})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{1}-\mu _{1})(X_{n}-\mu _{n})]\\\\\mathrm {E} [(X_{2}-\mu _{2})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{2}-\mu _{2})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{2}-\mu _{2})(X_{n}-\mu _{n})]\\\\\vdots &\vdots &\ddots
&\vdots \\\\\mathrm {E} [(X_{n}-\mu _{n})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{n}-\mu _{n})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{n}-\mu _{n})(X_{n}-\mu _{n})]\end{bmatrix}}}


矩阵中的第{\displaystyle (i,j)}

个元素是{\displaystyle
X_{i}}

与{\displaystyle
X_{j}}的协方差。这个概念是对于标量随机变数方差的一般化推广。