  A sequence of large invertible matrices given by a small random perturbation
around a fixed diagonal and positive matrix induces a random dynamics on a
high-dimensional sphere. For a certain class of rotationally invariant random
perturbations it is shown that the dynamics approaches the stable fixed points
of the unperturbed matrix up to errors even if the strength of the perturbation
is large compared to the relative increase of nearby diagonal entries of the
unperturbed matrix specifying the local hyperbolicity.
