Continuous Decompositions and Coalescing Eigenvalues for Matrices Depending on Parameters

Abstract

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem A(x) − λB(x), where A and B are symmetric matrix valued functions in R n×n , smoothly depending on parameters x ∈ Ω ⊂ R 2 ; further, B is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where A and B are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.

A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

“…By construction, these functions are holomorphic in the vicinity of ν * where eigenvalues coalesce, as this was already mentioned in [27,18,16,23] and in [4, p. 66]. The regularity of these two functions can easily be checked by injecting the local behavior given in Eq.…”

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

“…For hermitian problems, it can be shown that the veering phenomenon is a consequence of the eigenvalue problem degeneracy for a complex value of the parameter [31,38,42]. Indeed, if the parameter of the problem is extended in the complex-plane, the K (and M) matrices becomes non-Hermitian and eigenvalues and eigenvectors could coalesce.…”

“…As a consequence, for a given path in the complex parametric space, one can continuously change from one branch to the other. For instance, when ν move along the real axis, it explains the eigenvector switching at veering point [31]. Puiseux series are the only representation that allow capture the Riemann surface topology and to follow each modal branch.…”

“…Concurrently, the involved eigenvectors swap. This phenomenon is referred to as the veering phenomenon [30][31][32] in structural dynamics. In a stochastic context, the radius of convergence of perturbation methods is known to decrease when eigenvalues are getting closer to each other [9].…”

“…Although the literature has few criteria to qualify the veering intensity [33,34], the validity domain of perturbation methods for random eigenvalues is hardly predictable. The veering is known to be a consequence of the existence of branch point singularity [31], where eigenvalue coalesce, in the complex plane of the varying parameters referred to as Exceptional Point (EP) [20]. When the parameter is extended from real axis to the complex plane, the eigenvalue problem becomes non-Hermitian and this kind of system has recently attracted a large attention in classical and in Quantum Physics when losses, gain or Parity-Time symmetry [35] are introduced.…”