Introduction Matrix factorization has been proven to be one of the best ways to do collaborative filtering. The most common example of collaborative filtering is to predict how much a viewer will like a movie. The power of matrix factorization was a key development of the Netflix Prize (see http://www2.research.att.com/~volinsky/papers/ieeecomputer.pdf).
Using the movie rating example, the idea is that there are some underlying features of the movie and underlying attributes of the user that interact to determine if the user will like the movie.