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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