[benrbray]

Since you mentioned QR, I want to link [1] to one of my favorite articles, which explains how to use the QR decomposition to sample a random orthogonal matrix uniformly at random. The basic idea, of course, is that if A is a random matrix, and we compute A = QR, then Q is a random orthogonal matrix. But with what distribution? Well, it turns out to be almost uniform (according to Haar measure), but there are some density issues. A simple scaling of QR decomposition resolves the issue.

When I was a TA I loved giving this as an example to students. It's a great teaser for the rabbit hole that is random matrix theory.

[1] Mezzadri 2002, "How to Generate Random Matrices from the Classical Compact Groups" (http://www.ams.org/notices/200705/fea-mezzadri-web.pdf)