  . We study the evolution of the distribution of eigenvalues of a $N\times N$
matrix subject to a random perturbation drawn from (i) a generalized Gaussian
ensemble (ii) a non-Gaussian ensemble with a measure variable under the change
of basis. It turns out that, in the case (i), a redefinition of the parameter
governing the evolution leads to a Fokker-Planck equation similar to the one
obtained when the perturbation is taken from a standard Gaussian ensemble (with
invariant measure). This equivalence can therefore help us to obtain the
correlations for various physically-significant cases modeled by generalized
Gaussian ensembles by using the already known correlations for standard
Gaussian ensembles.
  For large $N$-values, our results for both cases (i) and (ii) are similar to
those obtained for Wigner-Dyson gas as well as for the perturbation taken from
a standard Gaussian ensemble. This seems to suggest the independence of
evolution, in thermodynamic limit, from the nature of perturbation involved as
well as the initial conditions and therefore universality of dynamics of the
eigenvalues of complex systems.
