Consider $Z = K - i \omega S - \omega^2 M$, where: $\omega$ is a positive real number; $K$ is a real, symmetric, and positive semi-definite matrix; $M$ is a real, symmetric, and positive definite matrix.
$S$ is a non-zero matrix.
Which property (or properties) $S$ should satisfy so that $Z$ is invertible? Or so that $(Z, M)$ has an eigendecomposition $Z = M \Phi \Lambda \Phi^T M$ (with $\Phi^T M \Phi = I$)?
Or which reference book discusses this class of problems?
Remark 1: Matrices like $Z$ arise in the discretization of the Helmholtz equation with absorbing boundary conditions or of the frequency response analysis with damping.
Remark 2: Many posts on the forum discuss the matrix $\left[ \begin{array}{cc} 1 & i \\ i & -1 \end{array} \right]$ as an example of non-diagonalizable complex symmetric matrix. Here I want to know which non-zero matrix $S$ would work.
EDIT: Generalization of proportional damping gives us a family of $S$ matrices that would allow the eigendecomposition: $S = M \sum_{j = J_1}^{J_2} s_j \left( M^{-1} K \right)^j$