Q: I'm looking for some code to do:
Fast Fourier Transforms (FFT's)
matrix arithmetic (multiplication, inversion, etc.)
A: Ajay Shah has prepared a nice index of free numerical software which has been archived pretty widely; one URL is ftp://ftp.math.psu.edu/pub/FAQ/numcomp-free-c . See also questions 18.9b, 18.13, 18.15c, and 18.16.