## 2 Background
### Spectral Transforms  
In general, the solution of the eigenvalue problem involves finding the full or partial spectrum of the pencil \(s\mathbf{E}-\mathbf{A}\). However, depending on the applied numerical method as well as on the structure of the system matrices, it is common that the eigenvalues are not found by directly using \(s\mathbf{E}-\mathbf{A}\), but through the pencil that arises from the application of a proper spectral transform. Spectral transforms are utilized by eigenvalue numerical methods usually for one of the following reasons:  
* To find the eigenvalues of interest. Some numerical methods, for example vector iteration-based methods, find the Largest Magnitude (LM) eigenvalues, whereas the eigenvalues of interest in SSSA are typically the ones with Smallest Magnitude (SM) or Largest real Part (LRP). Thus, it is necessary to apply a spectral transform, e.g., the invert or shift & invert. * Address a singularity issue. A GEP with singular left hand side coefficient matrix \(\mathbf{E}\) can create problems to many numerical methods. Applying a Mobius transform can help address singularity issues. * Accelerate convergence. Eigenvalues that are not very close to each other can lead to large errors and slow convergence of eigensolvers. Spectral transforms can help to magnify the eigenvalues of interest and speed up convergence. We describe here the Mobius transformation, which is a general variable transformation that includes as special cases all spectral transforms used in practice by eigenvalue algorithms. The formulation of the Mobius transformation is:  
\[s:=\frac{az+b}{cz+d}\,,\quad a,b,c,d\in\mathbb{C}\,,\quad ad-bc\
eq 0\,. \tag{9}\]  
The restriction in (9) is necessary, because, if \(ad=bc\), then \(s\) is constant which is not possible. Applying the transform (9) in (6) we obtain  
\[\det\!\left(\frac{az+b}{cz+d}\mathbf{E}-\mathbf{A}\right)=0\,,\]  
or, equivalently, by using determinant properties  
\[\det\!\left((az+b)\mathbf{E}-(cz+d)\mathbf{A}\right)=0\,,\]  
or, equivalently,  
\[\det\!\left((a\mathbf{E}-c\mathbf{A})z-(d\mathbf{A}-b\mathbf{E})\right)=0\,,\]  
which is the Characteristic equation characteristic equation of a linear dynamical system  
\[(a\mathbf{E}-c\mathbf{A})\dot{\mathbf{x}}(t)=(d\mathbf{A}-b\mathbf{E})\ddot{ \mathbf{x}}(t)\,, \tag{10}\]  
with pencil \(z(a\mathbf{E}-c\mathbf{A})-(d\mathbf{A}-b\mathbf{E})\). System (5) will be referred as the prime system, and the family of systems (10) will be defined as the proper \"M-systems\". An important property is that the solutions and stability properties of system (5) can be studied through (10) without resorting to any further computations, see [47]. The utilities of the family of systems of type (10) have been further emphasized by the features of some particular special cases. Table 1 summarizes the most commonly employed Mobius transforms and the corresponding matrix pencils for the GEP. The values of the parameters \(a,b,c,d\) that lead to each of these transforms are given in Table 2. In the case that \(\sigma>0\), the Cayley transform is equivalent to the bilinear transform \(z:=(\frac{T}{2}s+1)/(\frac{T}{2}s-1)\), where \(T=\frac{2}{\sigma}\). The choice of the best transform for a specific system and eigenvalue problem is a challenging task to solve, since the selection of shift values is always more or less heuristic. Finally, note that the importance of such spectral transforms for the eigenvalue analysis has also been identified in power system literature, see [48; 49].