I am asking for a follow up discussion to a question given by another user, based on the following URL link from this website asked about a year ago:


In particular, the answer provided a statement of a probability distribution but without much follow up discussion as to why that form of probability function arises, and without providing any references / sources for it (or a discussion on what the normalization factor might be).

To me it looks suspiciously close to the Matrix Langevin distribution (aka Matrix von-Mises Fisher), but I don't want to put the "cart before the horse", and instead am looking here for further clarification/justification.

And by extension (or at least to clarify the over-arching question, in case the linked question doesn't align with my ultimate question), suppose I had some $X=U\Sigma V^T$, where $U$, and $V$ were uniformly sampled over the Stiefel manifold according to Haar measure as outlined in Chikuse as implied for the following link:


Then what can be said about the distribution of $X$? (ignoring uniqueness issues). Intuitively it feels like it might follow something similar to the Matrix Langevin distribution (hence why I am interested with a follow up on this question / problem -- as it is difficult to find a textbook reference about this).



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