  We derive expressions for the neutrino mixing parameters that result from
complex perturbations on (1) the Majorana neutrino mass matrix (in the basis of
charged lepton mass eigenstates) and on (2) the charged lepton mass matrix, for
arbitrary initial (unperturbed) mixing matrices. In the first case, we find
that the phases of the elements of the perturbation matrix, and the initial
values of the Dirac and Majorana phases, strongly impact the leading order
corrections to the neutrino mixing parameters and phases. For experimentally
compatible scenarios wherein the initial neutrino mass matrix has $\mu-\tau$
symmetry, we find that the Dirac phase can take any value under small
perturbations. Similarly, in the second case, perturbations to the charged
lepton mass matrix can generate large corrections to the mixing angles and
phases of the PMNS matrix. As an illustration of our generalized procedure, we
apply it to a situation in which nonstandard scalar and nonstandard vector
interactions simultaneously affect neutrino oscillations.
