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When analyzing non-Hermitian lattice systems, the standard eigenmodedecomposition utilized for the analysis of Hermitian systems must be replacedby Jordan decomposition. This approach enables us to identify the correctnumber of the left and right eigenstates of a large finite-sized latticesystem, and to form a complete basis for calculating the resonant excitation ofthe system. Specifically, we derive the procedure for applying Jordandecomposition to a system with spectrally-isolated states. We use anon-Hermitian quadrupole insulator with zero-energy corner states as an exampleof a large system whose dimensionality can be drastically reduced to derive alow-dimensional "defective" Hamiltonian describing such localized states.Counter-intuitive and non-local properties of the resonant response of thesystem near zero energy are explained using the Jordan decomposition approach.Depending on the excitation properties of the corner states, we classify ournon-Hermitian quadrupolar insulator into three categories: trivial,near-Hermitian, and non-local.

Application of Jordan Decomposition to Non-Hermitian Lattice Models with Spectrally-Isolated Lower Dimensional States