The magnetic properties of the quiet Sun are investigated using a novel inversion code, FATIMA, based on the Principal Component Analysis of the observed Stokes profiles. The stability and relatively low noise sensitivity of this inversion procedure allows for the systematic inversion of large data sets with a very weak polarization signal. Its application to quiet-Sun observations of network and internetwork regions reveals that a significant fraction of the quiet-Sun contains kilogauss fields (usually with very small filling factors) and confirms that the pixels with weak polarization account for most of the magnetic flux. Mixed polarities in the resolution element are also found to occur more likely as the polarization weakens.