QEP with Cayley Transform.

With the so-called Cayley transform,

$$\mu=\frac{\alpha\lambda-\beta}{\lambda-\tau},$$

where the parameters $\alpha$, $\beta$, and $\tau$ are chosen such that $\alpha\tau-\beta\neq 1$, the original QEP (9.1) becomes

$$\left(\mu^2 \widehat{M} + \mu \widehat{C} + \widehat{K} \right) x = 0,$$ (262)

where $\widehat{M}=\tau^2 M+\tau C +K$, $\widehat{C}=-2\tau\beta M -(\alpha\tau+\beta)C-2\alpha K$, and $\widehat{K}=\beta^2 M +\alpha\beta C +\alpha^2 K$. Eigenvalues $\lambda$ of the original QEP (9.1) close to the antishift $\tau$ are transformed into large (in modulus) eigenvalues $\mu$ of the QEP (9.18). Eigenvalues $\lambda$ close to the shift $\beta/\alpha$ correspond to eigenvalues $\mu$ of (9.17) close to $0$.

Note that the triple $\{\widehat M,\widehat C,\widehat K\}$ is symmetric if that is the case for the real triple $\{ M,C,K \}$ and if $\alpha$, $\beta$, and $\tau$ are real.