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On the Conditional Distribution of a Multivariate Normal given a Transformation - the Linear Case.

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Rajeshwari Majumdar, Suman Majumdar

We show that the orthogonal projection operator onto the range of the adjointof a linear operator $T$ can be represented as $UT,$ where $U$ is an invertiblelinear operator. Using this representation we obtain a decomposition of aNormal random vector $Y$ as the sum of a linear transformation of $Y$ that isindependent of $TY$ and an affine transformation of $TY$. We then use thisdecomposition to prove that the conditional distribution of a Normal randomvector $Y$ given a linear transformation $\mathcal{T}Y$ is again a multivariateNormal distribution. This result is equivalent to the well-known result thatgiven a $k$-dimensional component of a $n$-dimensional Normal random vector,where $k