  We derive the Eigenstate Thermalization Hypothesis (ETH) from a random matrix
Hamiltonian by extending the model introduced by J. M. Deutsch [Phys. Rev. A
43, 2046 (1991)]. We approximate the coupling between a subsystem and a
many-body environment by means of a random Gaussian matrix. We show that a
common assumption in the analysis of quantum chaotic systems, namely the
treatment of eigenstates as independent random vectors, leads to inconsistent
results. However, a consistent approach to the ETH can be developed by
introducing an interaction between random wave-functions that arises as a
result of the orthonormality condition. This approach leads to a consistent
form for off-diagonal matrix elements of observables. From there we obtain the
scaling of time-averaged fluctuations with system size for which we calculate
an analytic form in terms of the Inverse Participation Ratio. The analytic
results are compared to exact diagonalizations of a quantum spin chain for
different physical observables in multiple parameter regimes.
