  In this paper, we present in-depth transfer matrix analyses of the Anderson
transition in three non-Hermitian (NH) systems, NH Anderson, U(1), and Peierls
models in three-dimensional systems. We first argue a general validity of the
transfer matrix analysis in NH systems, and clarify the symmetry properties of
the Lyapunov exponents, scattering ($S$) matrix and two-terminal conductance in
these NH models. The unitarity of the $S$ matrix is violated in NH systems,
where the two-terminal conductance can take arbitrarily large values.
Nonetheless, we show that the transposition symmetry of a Hamiltonian leads to
the symmetric nature of the $S$ matrix as well as the reciprocal symmetries of
the Lyapunov exponents and conductance in certain ways in these NH models.
Using the transfer matrix method, we construct a phase diagram of the NH
Anderson model for various complex single-particle energy $E$. We show that at
$E=0$, the phase diagram as well as critical properties become symmetric with
respect to an exchange of real and imaginary parts of on-site NH random
potentials. From finite size scaling analyses, we conclude that the exponents
of the NH class AI$^{\dagger}$ and the NH class A are $\nu=1.19 \pm 0.01$ and
$\nu=1.00 \pm 0.04$, respectively. In the NH models, a distribution of the
two-terminal conductance contains small fractions of huge conductance values,
which come from rare-event states with huge transmissions amplified by on-site
NH disorders. A geometric mean of the conductance enables a feasible
finite-size scaling analysis and the critical exponents obtained from the
conductance analysis are consistent with those from the localization length in
these three NH models.
