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Given the partitioning: $X = \begin{pmatrix}{} x_1 \\ \ x_2 \end{pmatrix}$ for a normal random vector with mean $\mu = \begin{pmatrix}{} \mu_1 \\ \ \mu_2 \end{pmatrix}$ and covariance matrix:

$\Sigma = \begin{pmatrix}{} \Sigma_{11} & \Sigma_{12} \\ \ \Sigma_{21} & \Sigma_{22} \end{pmatrix}$ where $\Sigma_{22}$ is invertible. How can I understand the following mean,covariance of a distribution of a random variable $X_1$ under condition $X_2 = x$? I'm struggling with formula for both the mean and covariance which are given specifically as:

$\mu_1 +\Sigma_{12}\Sigma_{22}^{-1}(x-\mu_2) $

and

$\Sigma_{11}- \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$.

I'm just getting into multivariate statistics, so I would preferably just try to get some intuition behind these two formulas if possible.

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    $\begingroup$ A great deal of intuition can come from studying the simplest (nontrivial) case: see our thread on this at stats.stackexchange.com/questions/71260. If that doesn't answer your question, could you please indicate what you might be looking for in addition? $\endgroup$
    – whuber
    May 12 '21 at 15:53
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    $\begingroup$ Yes, thank you. That solves my problem completely. $\endgroup$
    – PianoMath
    May 13 '21 at 10:58