Suppose we have two multivariate random variables $\mathbf{X}$ (of dimension $n_x$) and $\mathbf{Y}$ (of dimension $n_y$). The covariance matrix $C_{X,Y}$ can be written as the following block-matrix form: $$ \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \\ \end{bmatrix}, $$ where $\Sigma_{11}$ is the covariance of $\mathbf{X}$.

According to here, the conditional covariance matrix $C_{Y|X}$ can be expressed as:

$$ C_{Y|X}=\Sigma_{22}-\Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12} $$

My question is: how to derive the equality?


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