On the conditional distribution of a multivariate Normal given a transformation – the linear case

We show that the orthogonal projection operator onto the range of the adjoint [Formula: see text] of a linear operator T can be represented as UT, where U is an invertible linear operator. Given a Normal random vector Y and a linear operator T, we use this representation to obtain a linear operator [Formula: see text] such that [Formula: see text] is independent of TY and [Formula: see text] is an affine function of TY. We then use this decomposition to prove that the conditional distribution of a Normal random vector Y given [Formula: see text] , where [Formula: see text] is a linear transformation, 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 [Formula: see text] , the conditional distribution of the remaining [Formula: see text]-dimensional component is a [Formula: see text]-dimensional multivariate Normal distribution, and sets the stage for approximating the conditional distribution of Y given [Formula: see text] , where g is a continuously differentiable vector field.