  In the study of thermalization in finite isolated quantum systems, an
inescapable issue is the definition of temperature. We examine and compare
different possible ways of assigning temperatures to energies or equivalently
to eigenstates in such systems. A commonly used assignment of temperature in
the context of thermalization is based on the canonical energy-temperature
relationship, which depends only on energy eigenvalues and not on the structure
of eigenstates. For eigenstates, we consider defining temperature by minimizing
the distance between (full or reduced) eigenstate density matrices and
canonical density matrices. We show that for full eigenstates, the minimizing
temperature depends on the distance measure chosen and matches the canonical
temperature for the trace distance; however, the two matrices are not close.
With reduced density matrices, the minimizing temperature has fluctuations that
scale with subsystem and system size but appears to be independent of distance
measure. In particular limits, the two matrices become equivalent while the
temperature tends to the canonical temperature.
