  Three ways of constructing a non-Hermitian matrix with possible all real
eigenvalues are discussed. They are PT symmetry, pseudo-Hermiticity, and
generalized PT symmetry. Parameter counting is provided for each class. All
three classes of matrices have more real parameters than a Hermitian matrix
with the same dimension. The generalized PT-symmetric matrices are most general
among the three. All self-adjoint matrices process a generalized PT symmetry.
For a given matrix, it can be both PT-symmetric and P'-pseudo-Hermitian with
respect to some P' operators. The relation between corresponding P and P'
operators is established. The Jordan block structures of each class are
discussed. Explicit examples in 2x2 are shown.
