<p>The time-independent perturbation theory of Hermitian Hamiltonian in quantum mechanics is extended to the non-Hermitian Hamiltonian. The derivation procedure is given analytically and in detail. Some terms lost in literatures are added and an exact and complete non-Hermitian time-independent perturbation theory is obtained. The corrections of eigenenergies and eigenstates of two illustrative two-level non-Hermitian Hamiltonians, which have the Hermitian and non-Hermitian unperturbed Hamiltonian, respectively, are solved by another method, namely, the exactly-diagonalizing of the perturbed Hamiltonian in the Hilbert space of the unperturbed Hamiltonian. The results agree well with those from the non-Hermitian time-independent perturbation theory.</p>