Given a square matrix A, a Brauer’s theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt’s and Hotelling’s deflations. An extension of the aforementioned Brauer’s result, Rado’s theorem, shows how to modify r eigenvalues of A at the same time via a rank-r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem.
@article{bwmeta1.element.doi-10_2478_s11533-011-0113-0,
author = {Rafael Bru and Rafael Cant\'o and Ricardo Soto and Ana Urbano},
title = {A Brauer's theorem and related results},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {312-321},
zbl = {1248.15007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0113-0}
}
Rafael Bru; Rafael Cantó; Ricardo Soto; Ana Urbano. A Brauer’s theorem and related results. Open Mathematics, Tome 10 (2012) pp. 312-321. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0113-0/
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