  Mn$_3$O$_4$ is a spin frustrated magnet that adopts a tetragonally distorted
spinel structure at ambient conditions and a CaMn$_2$O$_4$-type postspinel
structure at high pressure. We conducted both optical measurements and
\emph{ab} \emph{initio} calculations, and systematically studied the electronic
band structures of both the spinel and postspinel Mn$_3$O$_4$ phases. For both
phases, theoretical electronic structures are consistent with the optical
absorption spectra, and display characteristic band-splitting of the conduction
band. The band gap obtained from the absorption spectra is 1.91(6) eV for the
spinel phase, and 0.94(2) eV for the postspinel phase. Both phases are
charge-transfer type insulators. The Mn 3\emph{d} $t_2$$_g$ and O 2\emph{p}
form antibonding orbitals situated at the conduction band with higher energy.
