On the Lidskii--Vishik--Lyusternik Perturbation Theory for Eigenvalues of Matrices with Arbitrary Jordan Structure

Abstract

Let A be a complex matrix with arbitrary Jordan structure and $\lambda$ 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 $\lambda$ under a small perturbation of A of order $\varepsilon$ is, generically, of order $\varepsilon^{1/n}$. 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 Hölder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors.