Yes, Golub/van Loan is the bible. But Walkins's "Fundamentals of Matrix Computations" (http://www.amazon.com/o/asin/0470528338/) is much more accessible to self-learners.
For those who are just starting on this subject, I highly recommend that you rework the proofs for the bounds between the various matrix norms [1]. These bounds are the building blocks for most interesting analyses in matrix computations. And understanding these analyses are essential if you want to know which algorithm will work best for your problems.
http://www.amazon.com/Computations-Hopkins-Studies-Mathemati...
This is the "how we wrote Lapack" book.