  Pseudo-hermitian matrices are matrices hermitian with respect to an
indefinite metric. They can be thought of as the truncation of pseudo-hermitian
operators, defined over some Krein space, together with the associated metric,
to a finite dimensional subspace. As such, they can be used, in the usual
spirit of random matrix theory, to model chaotic or disordered PT-symmetric
quantum systems, or their gain-loss-balanced classical analogs, in the phase of
broken PT-symmetry. The eigenvalues of pseudo-hermitian matrices are either
real, or come in complex-conjugate pairs. In this paper we introduce a family
of pseudo-hermitian random matrix models, depending parametrically on their
metric. We apply the diagrammatic method to obtain its averaged resolvent and
density of eigenvalues as explicit functions of the metric, in the limit of
large matrix size $N$. As a concrete example, which is essentially an ensemble
of elements of the non-compact unitary Lie algebra, we choose a particularly
simple set of metrics, and compute the resulting resolvent and density of
eigenvalues in closed form. The spectrum consists of a finite fraction of
complex eigenvalues, which occupy uniformly two two-dimensional blobs,
symmetric with respect to the real axis, as well as the complimentary fraction
of real eigenvalues, condensed in a finite segment, with a known non-uniform
density. The numbers of complex and real eigenvalues depend on the signature of
the metric, that is, the numbers of its positive and negative eigenvalues. We
have also carried thorough numerical analysis of the model for these particular
metrics. Our numerical results converge rapidly towards the asymptotic
analytical large-$N$ expressions.
