  We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian
is necessarily pseudo-Hermitian regardless of whether it is diagonalizable or
not. This result is different from Mostafazadeh's, which requires the
Hamiltonian to be diagonalizable. PT-symmetry breaking often occurs at
exceptional points where the Hamiltonian is not diagonalizable. Our result
implies that PT-symmetry breaking is equivalent to the onset of instabilities
of pseudo-Hermitian systems, which was systematically studied by Krein et al.
in 1950s. In particular, we show that the mechanism of PT-symmetry breaking is
the resonance between eigenmodes with different Krein signatures.
