  The Ricci Calabi functional is a functional on the space of K\"ahler metrics
of Fano manifolds. Its critical points are called generalized K\"ahler Einstein
metrics. In this article, we show that the Hessian of the Ricci Calabi
functional is non-negative at generalized K\"ahler Einstein metrics. As its
application, we give another proof of a Matsushima's type decomposition theorem
for holomorphic vector fields, which was originally proved by Mabuchi. We also
discuss a relation to the inverse Monge-Amp\`ere flow developed recently by
Collins-Hisamoto-Takahashi.
