Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. The implementation of the method in MATLAB code is available.Comment: 19 pages, 3 figure

Miniversal deformations of pairs of symmetric matrices under congruence

For each pair of complex symmetric matrices $(A,B)$ we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation that smoothly depends on the entries of $\widetilde{A}$ and $\widetilde{B}$. Such a normal form is called a miniversal deformation of $(A,B)$ under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from $(A,B)$ to its miniversal deformation.Comment: arXiv admin note: text overlap with arXiv:1104.249

Enhanced Spontaneous Emission at Third-Order Dirac Exceptional Points in Inverse-Designed Photonic Crystals

We formulate and exploit a computational inverse-design method based on topology optimization to demonstrate photonic crystal structures supporting complex spectral degeneracies. In particular, we discover photonic crystals exhibiting third-order Dirac points formed by the accidental degeneracy of monopolar, dipolar, and quadrupolar modes. We show that, under suitable conditions, these modes can coalesce and form a third-order exceptional point, leading to strong modifications in the spontaneous emission (SE) of emitters, related to the local density of states. We find that SE can be enhanced by a factor of 8 in passive structures, with larger enhancements ∼√n³ possible at exceptional points of higher order n.United States. Air Force Office of Scientific Research (FA9550-14-1-0389)National Science Foundation (U.S.) (DMR-1454836)National Science Foundation (U.S.) (DGE1144152

Reduction to versal deformations of matrix pencils and matrix pairs with application to control theory

Matrix pencils under the strict equivalence and matrix pairs under the state feedback equivalence are considered. It is known that a matrix pencil (or a matrix pair) smoothly dependent on parameters can be reduced locally to a special typically more simple form, called the versal deformation, by a smooth change of parameters and a strict equivalence (or feedback equivalence)transformation. We suggest an explicit recurrent procedure for finding the change of parameters and equivalence transformation in the reduction of a given family of matrix pencils (or matrix pairs) to the versal deformation. As an application, this procedure is applied to the analysis of the uncontrollability set in the space of parameters for a one-input linear dynamical system. Explicit formulae for a tangent plane to the uncontrollability set at its regular point and the perturbation of the uncontrollable mode are derived. A physical example is given and studied in detail

Transformation to versal deformations of matrices

AbstractIn the paper versal deformations of matrices are considered. The versal deformation is a matrix family inducing an arbitrary multi-parameter deformation of a given matrix by an appropriate smooth change of parameters and basis. Given a deformation of a matrix, it is suggested to find transformation functions (the change of parameters and the change of basis dependent on parameters) in the form of Taylor series. The general method of construction of recurrent procedures for calculation of coefficients in the Taylor expansions is developed and used for the cases of real and complex matrices, elements of classical Lie and Jordan algebras, and infinitesimally reversible matrices. Several examples are given and studied in detail. Applications of the suggested approach to problems of stability, singularity, and perturbation theories are discussed

Zur Stabilität von Systemen bewegter Kontinua mit Reibkontakten am Beispiel des Bremsenquietschens

Die vorliegende Arbeit befasst sich mit reibungsinduzierten Schwingungen bewegter elastischer Festkörper. Zunächst werden die Bewegungsgleichungen für verschiedene Kontaktformulierungen angegeben und nach einer Diskretisierung die zugehörigen linearisierten Störungsgleichungen ermittelt. Es folgen Betrachtungen zur Stabilität mit Fokus auf die zirkulatorischen Einflüsse. Schließlich wird ein analytisches Bremsenmodell hergeleitet und der Einfluß verschiedener Terme auf die Stabilität diskutiert