<think> We are given an N x k Gaussian matrix M (entries i.i.d. standard normal) with N >> k, and a diagonal matrix D = diag(λ₁, ..., λₙ). We are interested in the QR decomposition of DM, specifically the distribution of the Q factor (which is N x k with orthonormal columns).

In the case where k = N and D = I, Q is known to be uniformly distributed with respect to the Haar measure on the orthogonal group O(N). However, we are in the case k < N and with a general diagonal matrix D.

Let's first ...