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####

Conditioning.

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].

Susan Blackford
2000-11-20