Geodesic
Typical Changes in Spectral Properties under Perturbations by a Rank-One Operator
Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 590-602.

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We show that under perturbations by a rank-one typical operator, a fixed eigenvalue loses one Jordan block of maximal order, while the orders of the other Jordan blocks remain unchanged. We construct the first-order perturbation theory for the new eigenvalues and the zero-order approximations to the corresponding eigenvectors. 
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S. V. Savchenko. Typical Changes in Spectral Properties under Perturbations by a Rank-One Operator. Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 590-602. http://geodesic.mathdoc.fr/item/MZM_2003_74_4_a11/

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