Conditional distribution of multivariate normal

Question

$X \sim N(\mu, \Sigma)$. How can I find conditional distribution of $X$ given $AX$, where $A$ is a non-random matrix? I know that $AX \sim N(A\mu, A\Sigma A^T)$ but don't know what to do next.

P.S - I am preparing for an exam and solving problems from textbook, so should i add hashtag self-study?

I want to say it is not random. I am sorry for my bad english. — Markoff Chainz, Dec 07 '16 at 18:14
Are there some more assumptions on A e.g. invertible, full rank or something like this? — Gono, Dec 12 '16 at 13:09

score 1 · Accepted Answer · answered Dec 07 '16 at 18:30

$$ X \sim N(\mu, \Sigma) $$ \begin{align} \begin{bmatrix} X \\ Y \end{bmatrix} = \begin{bmatrix} X \\ AX \end{bmatrix} = \begin{bmatrix} I \\ A \end{bmatrix} X & = BX \sim N(B\mu, B\Sigma B^T) = N\left( \begin{bmatrix} \mu \\ A\mu \end{bmatrix} , \begin{bmatrix} \Sigma & \Sigma A^T \\ A\Sigma & A\Sigma A^T \end{bmatrix} \right) \\[10pt] & = N\left( \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} , \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{12}^T & \Sigma_{22} \end{bmatrix} \right) \end{align}

Do you know how to find the conditional distribution of $X$ given $Y$ when it's written in that form? (The conditional variance is a Schur complement.)

Yes I know that! Thank you very much! – Markoff Chainz Dec 07 '16 at 18:33 — Markoff Chainz, Dec 07 '16 at 18:33
@MarkoffChainz : I'm glad it helps. – Michael Hardy Dec 07 '16 at 18:39 — Michael Hardy, Dec 07 '16 at 18:39

Conditional distribution of multivariate normal

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