Shifted QEP.

With a shift $\lambda=\mu+\sigma$, the shifted QEP is

$$\left(\mu^2 \widehat{M} + \mu \widehat{C} + \widehat{K} \right) x = 0,$$ (260)

where $\widehat{M}=M$, $\widehat{C} = C + 2\sigma M$, and $\widehat{K} = K + \sigma C + \sigma^2 M$. The shift transforms eigenvalues $\lambda$ of (9.1) close to $\sigma$ into eigenvalues $\mu$ close to $0$.

The corresponding generalized ``linear'' eigenvalue problem is (again in terms of $\lambda$, rather than $\mu$)

$$\twobytwo{0}{I}{-\widehat{K}}{-\widehat{C}} \twobyone{x}{(\l... ...twobytwo{I}{0}{0}{\widehat{M}} \twobyone{x}{(\lambda-\sigma)x}$$

or

$$\twobytwo{0}{\widehat{K}}{\widehat{K}}{\widehat{C}} \twobyon... ...dehat{K}}{0}{0}{-\widehat{M}} \twobyone{x}{(\lambda-\sigma)x},$$

if the Hermitian of the matrix triplet $\{M,K,C\}$ wants to be preserved.