对一个多维随机变量作为线性变换以后的协方差矩阵

假设X=(x_{1},x_{2},...,x_{n})^{T}是一个n维的随机变量,它的协方差矩阵\Sigma_{X} =E\left [ (X-E(X))(X-E(X))^{T} \right ]

X做线性变换Y=AX,其中A是一个矩阵(当然也可以是一个标量),Y的协方差矩阵\Sigma _{Y}=A\Sigma _{X}A^{T}

证明如下:

\Sigma_{Y} =E\left [ (Y-E(Y)(Y-E(Y))^{T} \right ]

Y=AX代入\Sigma _{Y},得

\Sigma_{Y} =E\left [ (AX-E(AX))(AX-E(AX))^{T} \right ]

=E\left [ (AX-AE(X))(AX-AE(X))^{T} \right ]

=E\left [ A(X-E(X))(A(X-E(X)))^{T} \right ]

=E\left [ A(X-E(X))(X-E(X))^{T}A^{T} \right ]

=AE\left [ (X-E(X))(X-E(X))^{T} \right ]A^{T}

=A\Sigma _{X}A^{T}

你可能感兴趣的:(矩阵,线性代数)