Density of multivariate normal given linear condition Given a variable $X \sim N(\mu, \Sigma)$ 
What is the density of $X$ given $HX = d$,
$H$ and $d$ both constant
I.e. we observe $X$ with rank-deficient linear operator $H$ and obtain some value $d$. Given this knowledge how do we update our knowledge/distribution of $X$?
This is just one of the steps in the Kalman filter. I'm having trouble separating it out.
Ideally an answer would be analytic forms for both $\mu$ and $\Sigma$ after the update plus a rationale/derivation or link to such a derivation. 
 A: If you can accept that this b value has a variance, then the problem is solved.
The standard Kalman filter update step can handle non-square sensor-to-measurement matrices. You're already calling it 'H' so you're on the right track. 
You Just might need to include a finite variance on b or you'll end up with a degenerate (non-positive definite <--> one or more zero-Eigen-value) covariance matrix after the update step, and that may cause you problems elsewhere.
I bet that if you consider the variance of b to be zero the result will be identical to rotating the system so your slice is axis-aligned, and then running the Gaussian slice algorithm shown in Andrew Moore's data mining slides, and then rotating back to the original coordinate space. See the section on Gaussian  (http://www.autonlab.org/tutorials/gaussian.html).
If this isn't your only sensor just remember that the Kalman-filter update step is Bayesian inference (http://en.wikipedia.org/wiki/Bayesian_inference), and your simulation state is just a sensor. If your sensors errors are uncorrelated (which they should be or the basic Kalman filter won't be quite right) it doesn't matter what order you 'update' them together because a Bayseian update is commutative (http://en.wikipedia.org/wiki/Commutative_property) and associative (http://en.wikipedia.org/wiki/Associativity).
Sorry about the links, I don't have enough points to post more than two.
A: I suggest you to do this first. From the definition
http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Definition
the random vector $(X_1,\dots,X_k)$ is $N(\mu,\Sigma)$, where $\mu\in\mathbb{R}^k$ and $\Sigma$ is a $k\times k$ positive definite matrix with $\mathrm{rank}(\Sigma)=\ell$, if and only if $X=CZ+\mu$, where $Z_1,\dots,Z_k$ are independent $N(0,1)$, and $C$ is a $k\times \ell$ matrix such that $\Sigma = CC^\top$.
Use this to find the distribution of $HX = HCZ + H\mu$.
