Matrix Transformations

Consider the eigenvalue problem $Ax = \lambda Bx$. The spectral transformation or shift-and-invert transformation (SI) is defined by

$$T_{\rm SI} = (A - \mu B)^{-1} B,$$

where $\mu$ is the shift or pole. If $Ax = \lambda Bx$ then $T_{\rm SI} x = \gamma x$ with $\gamma=(\lambda-\mu)^{-1}$. An alternative is the Cayley transform

$$T_{\mathrm{C}} = (A-\mu B)^{-1} (A - \nu B),$$

where $\mu$ is the pole and $\nu$ the zero. If $Ax = \lambda Bx$ then $T_{\mathrm{C}} x=\zeta x$ with $\zeta = (\lambda-\mu)^{-1}(\lambda-\nu)$. Since $T_{\mathrm{C}} = I + (\mu-\nu) T_{\rm SI}$ and Krylov spaces are shift-invariant with respect to the matrix, we have that

$$\KK^k(T_{\mathrm{C}},v_1) = \KK^k(T_{\rm SI},v_1) \ ,$$

so, the Arnoldi method applied to $T_{\rm SI}$ or $T_{\mathrm{C}}$ delivers the same Ritz vectors and after back transformation of $\gamma$'s and $\zeta$'s, respectively, leads to the same $\lambda$'s.