  Hermitian operators with exact zero modes subject to non-Hermitian
perturbations are considered. Specific focus is on the average distribution of
the initial zero modes of the Hermitian operators. The broadening of these zero
modes is found to follow an elliptic Gaussian random matrix ensemble of fixed
size, where the symmetry class of the perturbation determines the behaviour of
the modes. This distribution follows from a central limit theorem of matrices,
and is shown to be robust to deformations of the average.
