The Eigenstate Thermalization Hypothesis (ETH) has been highly influential in
explaining thermodynamic behavior of closed quantum systems. As of yet, it is
unclear whether and how the ETH applies to non-Hermitian systems. Here, we
introduce a framework that extends the ETH to non-Hermitian systems. It hinges
on a suitable choice of basis composed of right eigenvectors of the
non-Hermitian model, a choice we motivate based on physical arguments. In this
basis, and after correctly accounting for the nonorthogonality of non-Hermitian
eigenvectors, expectation values of local operators reproduce the well-known
ETH prediction for Hermitian systems. We illustrate the validity of the
modified framework on non-Hermitian random-matrix and Sachdev--Ye--Kitaev
models. Our results thus generalize the ETH to the non-Hermitian setting, and
they illustrate the importance of the correct choice of basis to evaluate
physical properties.Comment: 5+5 pages, 3+3 figure