%0 Journal Article %T An explicit formula for the splitting of multiple eigenvalues for nonlinear eigenvalue problems and connections with the linearization for the delayeigenvalue problem %+ Department of Computer Science [Leuven] (DCS) %+ Dynamical Interconnected Systems in COmplex Environments (DISCO) %+ Laboratoire des signaux et systèmes (L2S) %+ Institut Polytechnique des Sciences Avancées (IPSA) %A Michiels, Wim %A Boussaada, Islam %A Niculescu, Silviu-Iulian %< avec comité de lecture %@ 0895-4798 %J SIAM Journal on Matrix Analysis and Applications %I Society for Industrial and Applied Mathematics %V 38 %N 2 %P 599–620 %8 2017-06-22 %D 2017 %R 10.1137/16M107774X %K nonlinear eigenvalue problems %K systems of functional equations and inequalities %K perturbations of nonlinear operators %K asymptotic distribution of eigenvalues and eigenfunctions %K matrix and operator equations %Z Engineering Sciences [physics]/Automatic %Z Mathematics [math]/Dynamical Systems [math.DS]Journal articles %X We contribute to the perturbation theory of nonlinear eigenvalue problems in three ways. First, we extend the formula for the sensitivity of a simple eigenvalue with respect to a variation of a parameter to the case of multiple nonsemisimple eigenvalues, thereby providing an explicit expression for the leading coefficients of the Puiseux series of the emanating branches of eigenvalues. Second, for a broad class of delay eigenvalue problems, the connection between the finite- dimensional nonlinear eigenvalue problem and an associated infinite-dimensional linear eigenvalue problem is emphasized in the developed perturbation theory. Finally, in contrast to existing work on analyzing multiple eigenvalues of delay systems, we develop all theory in a matrix framework, i.e., without reduction of a problem to the analysis of a scalar characteristic quasi-polynomial. %G English %2 https://centralesupelec.hal.science/hal-01558169/document %2 https://centralesupelec.hal.science/hal-01558169/file/MBN-SIAM-2017.pdf %L hal-01558169 %U https://centralesupelec.hal.science/hal-01558169 %~ CNRS %~ INRIA %~ UNIV-PSUD %~ INRIA-SACLAY %~ SUP_LSS %~ INSMI %~ INRIA_TEST %~ SUP_SYSTEMES %~ TESTALAIN1 %~ CENTRALESUPELEC %~ INRIA2 %~ TDS-MACS %~ UNIV-PARIS-SACLAY %~ UNIV-PSUD-SACLAY %~ INRIA-SACLAY-2015 %~ CENTRALESUPELEC-SACLAY %~ INRIA2017 %~ GS-ENGINEERING %~ GS-COMPUTER-SCIENCE %~ DISCO-L2S %~ INRIAARTDOI