Geodesic
On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 1, pp. 85-88.

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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
Mots-clés : Binet–Cauchy formula, Laurent series. 
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S. V. Savchenko. On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 1, pp. 85-88. http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a8/

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