Title: On the Conditional Distribution of a Multivariate Normal given a Transformation - the Linear Case,
Abstract: We show that the orthogonal projection operator onto the range of the adjoint
of a linear operator $T$ can be represented as $UT,$ where $U$ is an invertible
linear operator. Using this representation we obtain a decomposition of a
Normal random vector $Y$ as the sum of a linear transformation of $Y$ that is
independent of $TY$ and an affine transformation of $TY$. We then use this
decomposition to prove that the conditional distribution of a Normal random
vector $Y$ given a linear transformation $\mathcal{T}Y$ is again a multivariate
Normal distribution. This result is equivalent to the well-known result that
given a $k$-dimensional component of a $n$-dimensional Normal random vector,
where $k<n$, the conditional distribution of the remaining
$\left(n-k\right)$-dimensional component is a $\left(n-k\right)$-dimensional
multivariate Normal distribution, and sets the stage for approximating the
conditional distribution of $Y$ given $g\left(Y\right)$, where $g$ is a
continuously differentiable vector field.