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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