I've been reading this book lately, although I'm not finished yet.

It's basically a "second course" of matrix algebra that uses the full-rank factorization and the Moore-Penrose pseudoinverse to construct other generalized inverses and prove cool stuff about matrices. I initially borrowed a copy from the library for its extensive coverage of the Jordan decomposition (whose existence was really important for my control systems coursework), but I actually bought a copy as a reference because I found myself thumbing through it all the time. Although it is mostly theoretical, all the algorithms are covered sufficiently to do everything on paper if you wanted to.

If this isn't in the spirit of the community please let me know.