Let $w \sim N(\mu,\Sigma)$ be multivariate normal and let $w_1,w_2$ be two non-overlapping sub-vectors of $w$ respectively. Then how would $w$ be distributed conditional on the following statements $$S : = \{P_1w_1 = q_1,\quad P_2w_2 \sim N(q_2,\Omega)\}$$ in which $P_{1,2}$ are deterministic matrices and $q_{1,2}$ are deterministic vectors, $\Omega$ is the deterministic covariance matrix of $Pw_2$. Note that $P_{1,2}$ aren't assumed to be invertible or even square.
To give a quite generic example, let $w\in\Bbb R^6$, $w_1:=(w^{(1)}, w^{(2)})$, $w_2=(w^{(3)}, w^{(4)}, w^{(5)})$, and let the statements be: $$S:=\{w^{(1)}+w^{(2)}=2.5, \quad \begin{bmatrix} 1 & 0.5 & \sqrt{2}\\ -0.3 & 3.2 & 1.7 \end{bmatrix} w_2 \sim N( \begin{bmatrix} 0.2 \\ 0.3 \end{bmatrix}, \begin{bmatrix} 1 & 0\\ 0 & 2 \\ \end{bmatrix})\}$$ Then the question is to find $w\mid S$ or at least $\Bbb E(w\mid S)$. My guess is that $w\mid S$ is again normal, which is the case if the deterministic condition concerning $w_1$ is absent. Actually, my main problem is how to handle the deterministic part of $S$ or somewhow separate it from the probabilistic part.
EDIT: per @Did's comment, the non-deterministic part actually came from Black-Litterman's 1992 paper Global Portfolio Optimisation, in which the $\Omega$ measures how uncertain an investor is about his views on the returns of assets. In the Appendix part, the authors claimed that, with only the probabilistic condition, the conditional distribution of $w$ would be again a normal distribution.