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. The operator S is fixed and complex-valued. The unitary matrices U and V are drawn from the deformed Haar measure e z Re Tr[U V † ] dµ(U )dµ(V ), (I.4) where z > 0 is a fixed number setting the eccentricity of the limiting elliptic support of the level density. This freedom of alignment in the complex plane is missing in the first model where the spectrum is either elongated along the real or imaginary axis. Note that the Hermitian and the anti-Hermitian parts of this perturbation are now correlated. For z → ∞ and fixed matrix size, the two unitary matrices become almost the same U ≈ V .