  We propose an idea of eigenstate clustering in non-Hermitian systems. We show
that non-orthogonal eigenstates can be clustered around exceptional points and
illustrate our idea on some models. We discuss that exponential localization of
eigenstates at edges due to the non-Hermitian skin effect is a typical example
of eigenstate clustering. We numerically see that clustering of localized or
extended eigenstates are possible in systems with both open and closed
boundaries. We show that gain and loss can enhance eigenstate clustering. We
use fidelities and the standard k-means clustering algorithm for a systematic
study of clustered eigenstates.
