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