  We show that black holes grow substantially by accretion at close to
Eddington rates. Using a complete sample of soft X-ray selected AGNs,
  Grupe & Mathur (2004) have shown that narrow line Seyfert 1 galaxies, as a
class, lie below the M_BH-sigma relation of normal galaxies. Some NLS1s,
however, lie on or close to the M_BH-sigma relation. Here we show that not all
NLS1s accrete at close to Eddington rates: those with low {L/L_{Eddington}} are
close to the M_BH-sigma relation, and those with high {L/L_{Eddington}} are
far. With various tests in this paper, we argue that black holes grow in mass
substantially in their high-accretion phase and approach the M_BH-sigma
relation over time. The mass growth in a low accretion phase, as in BLS1s and
also in some NLS1s, appears to be insignificant. Any theoretical model
attempting to explain the M_BH-sigma relation will have to explain the above
observations.
