As stated before, eigenvalues of singular pencils
are discontinuous functions of the matrix entries.
For example, consider the singular pencil in
(2.5).
Changing to
and
to
makes this pencil regular, with an eigenvalue
at 1 and one at
, which can be arbitrary,
no matter how small the
values are.
Furthermore, changing
to
,
to
,
to
, and
to
,
leads to eigenvalues
and
,
both of which can be arbitrary, no matter how small the
values are.
Reducing subspaces can also change discontinuously.
Nonetheless, there are important situations where eigenvalues and reducing subspaces change continuously. This happens because the perturbations may be constrained to keep the dimension of a selected reducing subspace constant. In fact, the algorithms can be told to enforce such constraints when computing eigenvalues and reducing subspaces. This leads to useful bounds presented in detail in §8.7 and [119].