  We investigate the extent to which the eigenstate thermalization
hypothesis~(ETH) is valid or violated in the non-integrable and the integrable
spin-$1/2$ XXZ chain. We perform the energy-resolved analysis of the
statistical properties of matrix elements $\{O_{\gamma\alpha}\}$ of an
observable $\hat{O}$ in the energy eigenstate basis. The Hilbert space is
coarse-grained into energy shells of width $\Delta_E$, with which one can
define a block submatrix $\tilde{O}^{(b,a)}$ consisting of elements between
eigenstates in the $a$th and $b$th shells. Each block submatrix is
characterized by constant values of $E_{\gamma\alpha}=(E_\gamma+E_\alpha)/2
\simeq \tilde{E}$ and $\omega_{\gamma\alpha}= (E_\gamma-E_\alpha) \simeq
\omega$ up to $\Delta_E$. We will show that all matrix elements within a block
are statistically equivalent to each other in the non-integrable case. Their
distribution is characterized by $\bar{E}$ and $\omega$, and follows the
prediction of the ETH. In stark contrast, eigenstate-to-eigenstate fluctuations
persist in the integrable case. Consequently, matrix elements
$O_{\gamma\alpha}$ cannot be characterized by the energy parameters
$E_{\gamma\alpha}$ and $\omega_{\gamma\alpha}$ only. Our result explains the
origin for the breakdown of the fluctuation dissipation theorem in the
integrable system. The eigenstate-to-eigenstate fluctuations sheds a new light
on the meaning of the ETH.
