  There are some distinguished ensembles of non-Hermitian random matrices for
which the joint PDF can be written down explicitly, is unchanged by rotations,
and furthermore which have the property that the eigenvalues form a Pfaffian
point process. For these ensembles, in which the elements of the matrices are
either real, or real quaternion, the kernel of the Pfaffian is completely
determined by certain skew orthogonal polynomials, which permit an expression
in terms of averages over the characteristic polynomial, and the characteristic
polynomial multiplied by the trace. We use Schur polynomial theory, knowledge
of the value of a Schur polynomial averaged against real, and real quaternion
Gaussian matrices, and the Selberg integral to evaluate these averages.
