ABSTRACT Upcoming weak lensing surveys require a detailed theoretical understanding of the matter power spectrum in order to derive accurate and precise cosmological parameter values. While galaxy formation is known to play an important role, its precise effects are currently unknown. We present a set of 92 matter power spectra from the OWLS, cosmo-OWLS, and BAryons and HAloes of MAssive Systems simulation suites, including different ΛCDM cosmologies, neutrino masses, subgrid prescriptions, and AGN feedback strengths. We conduct a detailed investigation of the dependence of the relative difference between the total matter power spectra in hydrodynamical and collisionless simulations on the effectiveness of stellar and AGN feedback, cosmology, and redshift. The strength of AGN feedback can greatly affect the power on a range of scales, while a lack of stellar feedback can greatly increase the effectiveness of AGN feedback on large scales. We also examine differences in the initial conditions of hydrodynamic and N-body simulations that can lead to an $∼ 1{{\ \rm per\ cent}}$ discrepancy in the large-scale power, and furthermore show our results to be insensitive to cosmic variance. We present an empirical model capable of predicting the effect of galaxy formation on the matter power spectrum at z = 0 to within $1{{\ \rm per\ cent}}$ for $k\lt 1\, h\, \mathrm{Mpc}^{-1}$, given only the mean baryon fraction in galaxy groups. Differences in group baryon fractions can also explain the quantitative disagreement between predictions from the literature. All total and dark matter only power spectra in this library will be made publicly available at powerlib.strw.leidenuniv.nl.