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On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank

S. V. Savchenko

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We show that the $r$ largest Jordan blocks disappear and all other blocks remain the same in the part of the Jordan form corresponding to a given eigenvalue $\lambda$ under a generic rank $r$ perturbation. Moreover, a necessary and sufficient condition on the entries of a perturbation under which the spectral properties of $\lambda$ change in this manner is obtained with the use of the resolvent technique for the case in which the geometric multiplicity of $\lambda$ is greater than or equal to $r$. A Jordan basis in the corresponding root space is constructed from the Jordan chains of the original matrix. A complete description of how the spectrum changes in a small neighborhood of the point $z=\lambda$ is given for the case of a small parameter multiplying the perturbation.

Keywords: generic rank $r$ perturbation, scalar resolvent matrix, root space, Jordan block, Jordan basis, Binet–Cauchy formula, Laurent series.

UDC: 512.643+517.983

Received: 03.10.2002

DOI: 10.4213/faa100

 English version: Functional Analysis and Its Applications, 2004, 38:1, 69–71

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