15B57 Hermitian, skew-Hermitian, and related matrices
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- Popov function (1)
- Spectra of Graphs, Laplacians, Perturbations of Eigenvalues (1)
- behavior approach (1)
- contractivity (1)
- cyclo-dissipativity (1)
- dissipativity (1)
- even pencil (1)
- even polynomial (1)
- frequency-domain (1)
- generalized eigenvalue$ (1)
- generic perturbation (1)
- indefinite inner product (1)
- para-Hermitian polynomial (1)
- passivity (1)
- perturbation analysis (1)
- polynomial eigenvalue problem (1)
- problem (1)
- purely imaginary eigenvalue (1)
- rank one perturbation (1)
- selfadjoint matrices (1)
- structured eigenvalue problem (1)
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying
graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.
The behavior approach and the problem of dissipativity have both been introduced and studied extensively by Willems et al. However, a computationally feasible method to check dissipativity is missing. Current methods will mostly rely on symbolic representations of rational functions. We will discuss a new characterization for linear systems in behavior form that allows to check dissipativity via the solution of a para-Hermitian, polynomial eigenvalue problem. Thus, we can employ standard methods of cubic complexity.
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in
the sense of algebraic geometry. Special attention is paid to the perturbation
behavior of the sign characteristic. Typically, under such a perturbation,
for every given eigenvalue, the largest Jordan block of the eigenvalue is
destroyed and (in case the eigenvalue is real) all other Jordan blocks
keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of
the perturbed matrix that are not eigenvalues of the original matrix,
are typically simple, and in some cases information is provided about their sign
characteristic (if the new eigenvalue is real). The main results are proved by using
the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky
canonical form and on general results concerning rank one perturbations.