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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{FAA_2004_38_1_a8,
     author = {S. V. Savchenko},
     title = {On the {Change} in the {Spectral} {Properties} of a {Matrix} under {Perturbations} of {Sufficiently} {Low} {Rank}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {85--88},
     publisher = {mathdoc},
     volume = {38},
     number = {1},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a8/}
}
TY  - JOUR
AU  - S. V. Savchenko
TI  - On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2004
SP  - 85
EP  - 88
VL  - 38
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a8/
LA  - ru
ID  - FAA_2004_38_1_a8
ER  - 
%0 Journal Article
%A S. V. Savchenko
%T On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2004
%P 85-88
%V 38
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a8/
%G ru
%F FAA_2004_38_1_a8
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/