  This paper investigates finite-dimensional representations of PT-symmetric
Hamiltonians. In doing so, it clarifies some of the claims made in earlier
papers on PT-symmetric quantum mechanics. In particular, it is shown here that
there are two ways to extend real symmetric Hamiltonians into the complex
domain: (i) The usual approach is to generalize such Hamiltonians to include
complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real
symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the
first approach the spectrum remains real, while in the second approach the
spectrum remains real if the PT symmetry is not broken. Both generalizations
give a consistent theory of quantum mechanics, but if D>2, a D-dimensional
Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional
PT-symmetric matrix Hamiltonian.
