Do note that Golub & Van Loan is very much a reference text, however; it is not a great choice if you're just learning the subject (it covers everything, but without much depth and without much exposition).
True. J. W. Demmel, Applied Numerical Linear Algebra, is a gentler introduction.
However, if you're masochist enough to actually implement any of the algorithms, Golub & van Loan is a great reference. (Though, you really shouldn't implement it yourself except for didactic purposes - just use LAPACK/BLAS, which has been debugged for decades, and deals with all the special cases you're ignoring (underflow/overflow/nans/zeros/infs/...))
EDIT: Oh, and there's the excellent Numerical Linear Algebra by Trefethen and Bau, as kxyvr mentions.
EDIT EDIT: Funny, on amazon the top reviews for both books mentioned above are identical. Seems like I'm not the only one having trouble keeping them apart... :-) (Demmel is more of an introduction, FWIW)
I'd also recommend Fundamentals of Matrix Computations by Watkins for a great introduction. It helped immensely when I had to teach myself matrix algorithms as part of my undergraduate research.
If anyone wants an even more introductory book, this one (not yet published, but PDF drafts available) looks nice:
https://web.stanford.edu/~boyd/vmls/
Demmel is also, well, Applied. He goes into (some of the) the nitty-gritty implementation details, where Trefethen and Bau stay at a higher level of algorithmic abstraction.