  The eigenvalue PDF for some well known classes of non-Hermitian random
matrices --- the complex Ginibre ensemble for example --- can be interpreted as
the Boltzmann factor for one-component plasma systems in two-dimensional
domains. We address this theme in a systematic fashion, identifying the plasma
system for the Ginibre ensemble of non-Hermitian Gaussian random matrices $G$,
the spherical ensemble of the product of an inverse Ginibre matrix and a
Ginibre matrix $G_1^{-1} G_2$, and the ensemble formed by truncating unitary
matrices, as well as for products of such matrices. We do this when each has
either real, complex or real quaternion elements. One consequence of this
analogy is that the leading form of the eigenvalue density follows as a
corollary. Another is that the eigenvalue correlations must obey sum rules
known to characterise the plasma system, and this leads us to a exhibit an
integral identity satisfied by the two-particle correlation for real quaternion
matrices in the neighbourhood of the real axis. Further random matrix ensembles
investigated from this viewpoint are self dual non-Hermitian matrices, in which
a previous study has related to the one-component plasma system in a disk at
inverse temperature $\beta = 4$, and the ensemble formed by the single row and
column of quaternion elements from a member of the circular symplectic
ensemble.
