In a way motivated by the needs of the so called PT-symmetric
quantum mechanics of Bender et al and of Mostafazadeh et al (the
formalism of which is briefly summarized in section 2) it is shown,
via an exhaustive construction, that the N by N complex matrices
with real spectra are all P-pseudohermitian at N=2 (section 3) and
not all P-pseudohermitian at N=3 (section 4).

MR2659788 Wang, Qing-hai; Chia, Song-zhi; Zhang, Jie-hong $\scr{PT}$
symmetry as a generalization of Hermiticity. J. Phys. A 43 (2010),
no. 29, 295301, 11 pp. 81Q12