It is known that in various random matrix models, large perturbations create outlier eigenvalues which lie, asymptotically, in the complement of the support of the limiting spectral density. This thesis studies fluctuations of these outlier eigenvalues of iid matrices @math under bounded rank and bounded operator norm perturbations @math , namely the fluctuations @math . The perturbations @math that we consider belong to a large class, where we allow for arbitrary Jordan types and almost minimal assumptions on the left and right eigenvectors. We obtain the joint convergence of the normalized asymptotic fluctuations of the outlier eigenvalues in this setting with a unified approach.
 In this paper we study the distribution of eigenvalues for two sets of random Hermitian matrices and one set of random unitary matrices. The statement of the problem as well as its method of investigation go back originally to the work of Dyson [i] and I. M. Lifsic [2], [3] on the energy spectra of disordered systems, although in their probability character our sets are more similar to sets studied by Wigner [4]. Since the approaches to the sets we consider are the same, we present in detail only the most typical case. The corresponding results for the other two cases are presented without proof in the last section of the paper. §1. Statement of the problem and survey of results We shall consider as acting in iV-dimensiona l unitary space v, a selfadjoint operator BN (re) of the form Let A be a complex matrix with arbitrary Jordan structure and @math an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S. R. Comput. Math. and Math. Phys., 1 (1965), pp. 73--85], showing that the splitting of @math under a small perturbation of A of order @math is, generically, of order @math . Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Holder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors. This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type @math , where the rank of @math stays bounded as the dimension goes to infinity and where the matrix @math is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if @math has some eigenvalues out of the maximal circle of the single ring, then @math has some eigenvalues (called outliers) in the neighborhood of those of @math , which is not the case for the eigenvalues of @math in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of @math around the eigenvalues of @math and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of @math ) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but only in the case of non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.