[FabHK]

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)

[hcho3]

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.

https://www.amazon.com/Fundamentals-Matrix-Computations-Davi...

[jacobolus]

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/

[stephencanon]

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.