  Observations have shown that supermassive black holes in nearby elliptical
galaxies correlate tightly with the stellar velocity dispersion (the $\MBH -
\sigma$ relation) and the stellar mass (the $\MBH - \Mhost$ relation) of their
host spheroids. However, the origin of these correlations remains ambiguous. In
a previous paper by Zhu et al., we proposed a model which links the M-$\sigma$
relation to the the dynamical state of the system and the $\MBH - \Mhost$
relation to the self-regulation of galaxy growth. To test this model, we
compile a sample of observed galaxies with different properties and examine the
dependence of the above correlations on these parameters. We find that galaxies
that satisfy the the $\MBH - \sigma$ correlation appear to have reached virial
equilibrium, as indicated by the ratio between kinetic energy and gravitational
potential, 2K/U $\sim$ 1. Furthermore, the ratio of black hole accretion rate
to star formation rate remains nearly constant, BHAR /SFR $\sim$ $10^{-3}$, in
active galaxies over a wide range of mass in the redshift range z=0 - 3. These
results confirm our theoretical model that the observed correlations have
different origins: the $\MBH - \sigma$ relation may result from galaxy
relaxation, while the $\MBH$ - $\Mhost$ relation may be due to self-regulated
black hole accretion and star formation in galaxies.
