To determine the probability density function (PDF) of the "unitary" matrix ( Q ) from the QR decomposition of ( DM ), where ( M ) is an ( N \times k ) Gaussian matrix with i.i.d. standard normal entries and ( D = \text{diag}(\lambda_1, \ldots, \lambda_N) ) is a diagonal matrix, we analyze the effect of row scaling on the distribution of ( Q ).

Key Observations:

1. **QR Decomposit...

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...

QR Decomposition of Normally Distributed Matrices: Distribution Analysis

This question deals with finding the probability distribution function (PDF) of the unitary matrix Q obtained from the QR decomposition of scaled Gaussian matrices.

For the k=1 case (simplest case)

When k=1, the QR decomposition simply normalizes the vector. If M is an N-dimensional random vector with i.i.d. standard ...

This is an interesting question about the distribution of Q matrices from QR decompositions of scaled Gaussian matrices. Let me provide some insights on the different cases.

General Framework

When is an matrix with i.i.d. standard normal entries and we compute the QR decomposition of , we have: where is with orthonormal columns and is $k ...

Notation
• M – N × k matrix whose entries are i.i.d. 𝒩(0,1) (N ≥ k)
• D = diag(λ₁,…,λ_N) with all λ_i > 0
• Y := D M, and let

  Y = Q R ,  Q ∈ V_{N,k}(ℝ) ,  R upper–triangular with r_{jj}>0 (1)

(Q has orthonormal columns, i.e. QᵀQ = I_k; V_{N,k}(ℝ) is the (real) Stiefel manifold).

Goal – find the probability density of Q.

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Below is a fairly complete description of the law of the “orientation”‐factor (Q) in the economy QR‐factorization [ DM ;=;Q,R,\quad M\sim N_{N\times k}(0,I_N\otimes I_k),;D=\diag(\lambda_1,\dots,\lambda_N)>0, ] where (N\gg k). One shows by the usual change-of-variables (M\mapsto(Q,R)) and then integrating out the upper-triangular (R) that the marginal density of (Q) on the Stiefe...

In the case where k = N and D = I, Q is known to be uniformly distributed with respect to the Haar measure on the orthogon...

Excellent question. This delves into the fascinating area of random matrix theory and directional statistics. Let's break down the problem, starting from the simplest case and building up to the general one.

Summary of the Problem

You have a matrix , where:

• is an matrix with i.i.d. entries.
• $D = \text{diag}(\lambda_1, \dots, \la...

Let

  M = (Mij) ∈ ℝN×k ,  Mij i.i.d. N(0,1),   N ≥ k ,
  D = diag(λ1 ,…, λN )  with λi > 0 ,

and write

  Y = D M. (1)

Put the (thin) QR–decomposition of Y

  Y = Q R , Q ∈ 𝒱N,k (QᵀQ = Ik),  R ∈ ℝk×k upper-triangular, R11,…,Rkk > 0, (2)

where 𝒱N,k is the Stiefel manifold of N × k orthonormal frame...