How does parity-time symmetry break? -- pseudo-Hermiticity and resonance between positive- and negative-action modes

It is generally believe that Parity-Time(PT)-symmetry breaks at exceptional points (EPs). However, we prove that EPs are necessary but not sufficient conditions for PT-symmetry breaking. We show that PT-symmetry breaks at EPs when a postive-action mode resonates with a negative-action mode, and does not at EPs when the resonance is between two modes with the same sign of actions. This is accomplished by proving that in finite dimensions, a PT-symmetric Hamiltonian is necessarily pseduo-Hermitian, regardless whether it is diagonalizable or not. It is shown that PT-symmetry can break at EPs where the Hamiltonian can be either diagonalizable or non-diagonalizable.