  Understanding the evolution towards thermal equilibrium of an isolated
quantum system is at the foundation of statistical mechanics and a subject of
interest in such diverse areas as cold atom physics or the quantum mechanics of
black holes. Since a pure state can never evolve into a thermal density matrix,
the Eigenstate Thermalization Hypothesis (ETH) has been put forward by Deutsch
and Srednicki as a way to explain this apparent thermalization, similarly to
what the ergodic theorem does in classical mechanics. In this paper this
hypothesis is tested numerically. First, it is observed that thermalization
happens in a subspace of states (the Krylov subspace) with dimension much
smaller than that of the total Hilbert space. We check numerically the validity
of ETH in such a subspace, for a system of hard core bosons on a
two-dimensional lattice. We then discuss how well the eigenstates of the
Hamiltonian projected on the Krylov subspace represent the true eigenstates.
This discussion is aided by bringing the projected Hamiltonian to the
tridiagonal form and interpreting it as an Anderson localization problem for a
finite one-dimensional chain. We also consider thermalization of a subsystem
and argue that generation of a large entanglement entropy can lead to a thermal
density matrix for the subsystem well before the whole system thermalizes.
Finally, we comment on possible implications of ETH in quantum gravity.
