  Taking into account the density matrices with non-full ranks, we show that
the fidelity susceptibility is determined by the support of the density matrix.
Combining with the corresponding expression of the quantum Fisher information,
we rigorously prove that the fidelity susceptibility is proportional to the
quantum Fisher information. As this proof can be naturally extended to the full
rank case, this proportional relation is generally established for density
matrices with arbitrary ranks. Furthermore, we give an analytical expression of
the quantum Fisher information matrix, and show that the quantum Fisher
information matrix can also be represented in the density matrix's support.
