Let X = D M, where M is N×k with i.i.d. N(0,1) entries and D = diag(λ1,…,λN) is fixed. Then X has a matrix normal distribution X ∼ MNN×k(0, Σ, Ik) with row covariance Σ = D^2 and column covariance Ik: vec(X) ∼ N(0, Ik ⊗ Σ).

Write the QR (more conveniently, polar) decomposition of X as X = Q R, with Q ∈ VNk (the Stiefel manifold of N×k matrices with orthonormal columns) and R upper triangular with positive diagonal. Equivalently, let T = R R′ ∈ Sym+ k and write X = Q T1/2.

Distribution of Q (general N,k and general D) Using the change-of-variables formula for the QR/polar decomposition (e.g., Muirhead, Aspects of Multivariate Statistical Theory; Chikuse, Statistics on Special Manifolds), the Lebesgue element transforms as dX = 2−k |T|(N−k−1)/2 dT dμ(Q), where dμ(Q) is the (Riemannian/Haar) invariant measure on VNk.

The density of X is fX(X) ∝ |Σ|−k/2 exp{−(1/2) tr(Σ−1 X X′)}. With X = Q T1/2 this becomes tr(Σ−1 X X′) = tr(Σ−1 Q T Q′) = tr(T A), where A := Q′ Σ−1 Q.

Integrating out T using the standard Wishart integral ∫T>0 exp{−(1/2) tr(A T)} |T|(N−k−1)/2 dT ∝ |A|−N/2 yields the marginal density of Q (with respect to the uniform probability measure on VNk): fQ(Q) = |Σ|−k/2 |Q′ Σ−1 Q|−N/2. This distribution is known as the matrix angular central Gaussian (MACG) distribution on the Stiefel manifold with parameter Σ (defined up to a positive scalar multiple). It reduces to the uniform (Haar) distribution on VNk when Σ is proportional to the identity, and for k = N it reduces to Haar on O(N) regardless of Σ (since |Q′ Σ−1 Q| = |Σ−1|).

Special cases and remarks

• k = 1 (unit sphere): Q is just a unit vector u ∈ SN−1, and f(u) = |Σ|−1/2 (u′ Σ−1 u)−N/2, the angular central Gaussian (ACG) distribution. Taking Σ = D^2 = diag(λ1^2,…,λN^2) gives f(u) ∝ (∑i u_i^2 / λ_i^2)−N/2. When all λi are equal, this is constant (uniform on the sphere), reproducing the result you cited.

• General diagonal D and D^2 M, etc.: If you replace X by D^α M for any α > 0, then Σ = D2α and the same formula applies with that Σ. The distribution depends on Σ only up to an overall positive scalar (as expected, since scaling X by a scalar does not change its Q).

• Dependence on subspace only: The density depends on Q only through the k-dimensional subspace it spans, since |(QO)′ Σ−1 (QO)| = |Q′ Σ−1 Q| for any O ∈ O(k). Thus the induced distribution on the Grassmann manifold of k-planes has density proportional to |Q′ Σ−1 Q|−N/2 as well.

• Independence: Q and R are independent if and only if Σ is proportional to the identity. Otherwise they are dependent, and Q is not uniformly distributed.

References

• H. Chikuse (2003). Statistics on Special Manifolds. Springer. See chapters on Stiefel/Grassmann manifolds and the matrix angular central Gaussian distribution (MACG).
• R. J. Muirhead (1982). Aspects of Multivariate Statistical Theory. Wiley. See change-of-variables for QR/polar decomposition and Wishart integrals.
• D. E. Tyler (1987). A distribution-free M-estimator of multivariate scatter. Annals of Statistics. (For the vector ACG case.)
• H. Chikuse (1998). The matrix angular central Gaussian distribution. Journal of Multivariate Analysis 66: 251–270.

In summary: If X = D M with Σ = D^2, then the Q from the QR decomposition of X has density on the Stiefel manifold f(Q) = |D^2|−k/2 |Q′ D−2 Q|−N/2 with respect to the uniform probability measure. For k = 1 this is exactly the angular central Gaussian distribution; for Σ ∝ I it reduces to the uniform (Haar) distribution.