## 2 Background
### Eigenvalue Problem  
In this paper, we are concerned with finding the eigenvalues of the following system:  
\[\mathbf{E}\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)\,, \tag{5}\]  
where \(\mathbf{E},\mathbf{A}\in\mathbb{R}^{r\times r},\mathbf{x}:[0,+\infty]\mapsto \mathbb{C}^{r\times 1}\). Matrix \(\mathbf{E}\) in (5) can be:  
* non-singular, i.e., \(\det(\mathbf{E})\
eq 0\). This is the case of the ODE power system model (4). In particular, (4) can be obtained from (5) for \(r=n\), \(\mathbf{x}\equiv\Delta\mathbf{x}\), \(\mathbf{A}\equiv\mathbf{A}_{\rm S}\), \(\mathbf{E}\equiv\mathbf{I}_{r}\).
* singular, i.e., \(\det(\mathbf{E})=0\). This is the case of the DAE power system model (2). In particular, (2) can be obtained from (5) for \(r=n+m\), \(\mathbf{x}\equiv\mathbf{z}\), \(\mathbf{A}\equiv\mathbf{A}_{\rm I}\), \(\mathbf{E}\equiv\mathbf{E}_{\rm I}\).  
The stability of (5) can be assessed by calculating its eigenvalues, which are defined as the roots of the characteristic equation:  
\[\det(s\mathbf{E}-\mathbf{A})=0\,, \tag{6}\]  
where \(s\in\mathbb{C}\) denotes the complex Laplace variable. In (6), \(s\mathbf{E}-\mathbf{A}\) is called the matrix pencil and \(\det(s\mathbf{E}-\mathbf{A})\) the _characteristic polynomial_ of system (5). In this paper, we assume that \(\det(s\mathbf{E}-\mathbf{A})=\Pi(s)\
ot\equiv 0\), where \(\Pi(s)\) is polynomial of order equal or less than \(r\), in which case \(s\mathbf{E}-\mathbf{A}\) is called a regular pencil [46]. In general, analytical solution of (6) is possible only if \(r\leq 4\). For higher degrees, general formulas do not exist and only the application of a numerical method is possible. In addition, algorithms that explicitly determine the characteristic polynomial \(\det(s\mathbf{E}-\mathbf{A})\) and then numerically calculate its roots, may be extremely slow, even for small problems. Alternatively, the eigenvalues of \(s\mathbf{E}-\mathbf{A}\) can be found from the solution of the Generalized Eigenvalue Problem (GEP):  
\[\begin{split}(s\mathbf{E}-\mathbf{A})\mathbf{u}&= \mathbf{0}_{r,1}\,,\\ \mathbf{w}(s\mathbf{E}-\mathbf{A})&=\mathbf{0}_{1,r} \,,\end{split} \tag{7}\]  
where \(\mathbf{u}\in\mathbb{C}^{r\times 1}\) and \(\mathbf{w}\in\mathbb{C}^{1\times r}\). Every value of \(s\) that satisfies (7) is an eigenvalue of the pencil \(s\mathbf{E}-\mathbf{A}\), with the vectors \(\mathbf{u}\), \(\mathbf{w}\) being the corresponding right and left eigenvectors, respectively. Thus, the solution of the GEP consists in calculating the eigenpairs, i.e., eigenvalues and eigenvectors, which satisfy (7). It may be required that only right (or left) or both right and left eigenvectors are calculated, depending on the analysis that needs to be carried out. In general, the pencil \(s\mathbf{E}-\mathbf{A}\) has \(\mathrm{rank}(s\mathbf{E}-\mathbf{A})\) finite eigenvalues and the infinite eigenvalue with multiplicity \(r-\mathrm{rank}(s\mathbf{E}-\mathbf{A})\). Note that, if \(\mathbf{E}\) is singular, then the pencil will have the infinite eigenvalue with multiplicity of at least one.  
In the special case that the left-hand side matrix of (5) is the identity matrix, e.g., (4), the general problem (7) is reduced to the following Linear Eigenvalue Problem (LEP):  
\[\begin{split}(s\mathbf{I}_{r}-\mathbf{A})\mathbf{u}& =\mathbf{0}_{r,1}\,,\\ \mathbf{w}(s\mathbf{I}_{r}-\mathbf{A})&=\mathbf{0}_{1,r}\,.\end{split} \tag{8}\]  
the solution of the LEP consists in calculating the \(r\) finite eigenvalues and eigenvectors of \(s\mathbf{I}_{r}-\mathbf{A}\).