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Quantum physics is generally concerned with real eigenvalues due to theunitarity of time evolution. With the introduction of $\mathcal{PT}$ symmetry,a widely accepted consensus is that, even if the Hamiltonian of the system isnot Hermitian, the eigenvalues can still be pure real under specific symmetry.Hence, great enthusiasm has been devoted to exploring the eigenvalue problem ofnon-Hermitian systems. In this work, from a distinct perspective, wedemonstrate that real eigenvalues can also emerge under the appropriaterecursive condition of eigenstates. Consequently, our findings provide anotherpath to extract the real energy spectrum of non-Hermitian systems, whichguarantees the conservation of probability and stimulates future experimentalobservations.

Real eigenvalues are determined by the recursion of eigenstates