  We present an eigenfunction method to analyze 161 visual light curves (LCs)
of Type Ia supernovae (SNe Ia) obtained by the Carnegie Supernova Project to
characterize their diversity and host-galaxy correlations. The eigenfunctions
are based on the delayed-detonation scenario using three parameters: the LC
stretch being determined by the amount of deflagration-burning governing the
56Ni production, the main-sequence mass M_MS of the progenitor white dwarf
controlling the explosion energy, and its central density rho_c shifting the
56Ni distribution. Our analysis tool (SPAT) extracts the parameters from
observations and projects them into physical space using their allowed ranges
M_MS < 8 M_sun, rho_c < 7-8x10^9g/cc. The residuals between fits and individual
LC-points are ~ 1-3% for ~ 92% of objects. We find two distinct M_MS groups
corresponding to a fast (~ 40-65 Myrs) and a slow(~ 200-500 Myrs) stellar
evolution. Most underluminous SNe Ia have hosts with low star formation but
high M_MS, suggesting slow evolution times of the progenitor system. 91T-likes
SNe show very similar LCs and high M_MS and are correlated to star formation
regions, making them potentially important tracers of star formation in the
early Universe out to z = 4-11. Some 6% outliers with `non-physical' parameters
can be attributed to superluminous SNe Ia and subluminous SNe Ia with hosts of
active star formation. For deciphering the SNe Ia diversity and high-precision
SNe Ia cosmology, the importance is shown for LCs covering out to ~ 60 days
past maximum. Finally, our method and results are discussed within the
framework of multiple explosion scenarios, and in light of upcoming surveys.
