  We elaborate on the deviation of the Jordan structures of two linear
relations that are finite-dimensional perturbations of each other. We compare
their number of Jordan chains of length at least $n$. In the operator case, it
was recently proved that the difference of these numbers is independent of $n$
and is at most the defect between the operators. One of the main results of
this paper shows that in the case of linear relations this number has to be
multiplied by $n+1$ and that this bound is sharp. The reason for this behavior
is the existence of singular chains. We apply our results to one-dimensional
perturbations of singular and regular matrix pencils. This is done by
representing matrix pencils via linear relations. This technique allows for
both proving known results for regular pencils as well as new results for
singular ones.

Functional Analysis