We learned about the QR factorization in our class. The QR factorization has so many uses that the search for accurate and faster QR factorization algorithms goes on. We could use the QR factorization in the eigenvalue and linear least squares problem to re arrange the columns of the matrix in order of linear dependence using the column pivoting (QRP) algorithm. And we could just remove the linear dependent columns to get the ones which span the basis of the matrix given quickly.

A team of researchers used QR factorization to produce a small basis of protein structures which would best span the evolutionary (phase) space. They encoded their data in a matrix of dimension m*n*d where m is the length of the multiple alignments while n is the number of proteins in the alignment. d stands for the set of rotated coordinates of each protein in the different alignments. Each column of their matrix corresponded to a single protein structure. Since the QRP algorithm runs for 2D matrices, they basically just run the QRP algorithm simultaneously for d matrices each of size m*n.
Then the QRP algorithm was run to arrange all the protein molecules in the order of increasing linear dependence. Thus, they discarded the first few columns and then obtained their representative molecule profiles.

They got better results than the molecules obtained by homology detection searches over large databases. They presented a measure of likeness between similar molecules and then made an evolutionary phylogenetic tree out of the information they received which they found out was similar to the max likelihood sequence-based phylogeny. And thus they were able to verify that the evolutionary information could be found out from the analysis of the protein structures itself.

References:
http://www.netlib.org/lapack/lug/node42.html
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK7-4DXT3X4-H&_user=492137&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000022719&_version=1&_urlVersion=0&_userid=492137&md5=4ff9daf450dfe5227f84b89be9ac966b#sec7