We point out that in non-Hermitian systems, Jordan decomposition should replace eigendecomposition so that all “good approximate eigenstates” of the system are identified. These states can be resonantly excited. As a concrete example, we study the location and field distribution of zero-energy corner states in a non-Hermitian quadrupole insulator (QI) and split the parameter space into three distinct regimes according to properties of the corner states: near-Hermitian QI, intermediate phase, and trivial insulator. In the newly discovered intermediate phase, the Hamiltonian becomes defective, and the counterintuitive and delicate response of the system to external drives can be perfectly explained by the Jordan decomposition.

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• Symmetry protected topological states
• Topological phase transition
• Nonequilibrium systems
• Topological materials