In numerical linear algebra, we know how Householder QR can handle rank deficient matrices with column pivoting: which is essentially to choose the left-over columns with the maximum norm and use permutation matrices to put it in front, for instance, see page 18-19 of:
https://www-users.cse.umn.edu/~saad/csci5304/FILES/LecN8.pdf
Yet, little were known about the justification of this method. I was told that it is discovered and used simply because it worked for the most of the time. I was also told that there are counter-examples where this method fails. Little literature can I find for Householder QR with column pivoting in general not to mention the case when it fails. Could somebody provide some justification and literature on this?