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Matrix Transformations
Consider the eigenvalue problem
.
The spectral transformation or shift-and-invert transformation (SI)
is defined by
where
is the shift or pole.
If
then
with
.
An alternative is the Cayley transform
where
is the pole and
the zero.
If
then
with
.
Since
and Krylov spaces are
shift-invariant
with respect to the matrix, we have that
so, the Arnoldi method applied to
or
delivers the same
Ritz vectors and after back transformation of
's and
's,
respectively, leads to the same
's.
Susan Blackford
2000-11-20