A Brauer’s theorem and related results

Rafael Bru; Rafael Cantó; Ricardo Soto; Ana Urbano

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 312-321
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Rafael Bru, et al. "A Brauer’s theorem and related results." Open Mathematics 10.1 (2012): 312-321. <http://eudml.org/doc/269196>.

@article{RafaelBru2012,
abstract = {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.},
author = {Rafael Bru, Rafael Cantó, Ricardo Soto, Ana Urbano},
journal = {Open Mathematics},
keywords = {Eigenvalues; Pole assignment problem; Controllability; Low rank perturbation; Deflation techniques; eigenvalues; pole assignment problem; controllability; low rank perturbation; deflation techniques; stabilization; Jordan form; Wielandt's and Hotelling's deflations},
language = {eng},
number = {1},
pages = {312-321},
title = {A Brauer’s theorem and related results},
url = {http://eudml.org/doc/269196},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Rafael Bru
AU - Rafael Cantó
AU - Ricardo Soto
AU - Ana Urbano
TI - A Brauer’s theorem and related results
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 312
EP - 321
AB - 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.
LA - eng
KW - Eigenvalues; Pole assignment problem; Controllability; Low rank perturbation; Deflation techniques; eigenvalues; pole assignment problem; controllability; low rank perturbation; deflation techniques; stabilization; Jordan form; Wielandt's and Hotelling's deflations
UR - http://eudml.org/doc/269196
ER -

References

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  1. [1] Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75–91 http://dx.doi.org/10.1215/S0012-7094-52-01910-8 Zbl0046.01202
  2. [2] Crouch P.E., Introduction to Mathematical Systems Theory, Mathematik-Arbeitspapiere, Bremen, 1988 
  3. [3] Delchamps D.F., State-Space and Input-Output Linear Systems, Springer, New York, 1988 http://dx.doi.org/10.1007/978-1-4612-3816-4 
  4. [4] Hautus M.L.J., Controllability and observability condition of linear autonomous systems, Nederl. Akad. Wetensch. Indag. Math., 1969, 72, 443–448 Zbl0188.46801
  5. [5] Kailath T., Linear Systems, Prentice Hall Inform. System Sci. Ser., Prentice Hall, Englewood Cliffs, 1980 
  6. [6] Langville A.N., Meyer C.D., Deeper inside PageRank, Internet Math., 2004, 1(3), 335–380 http://dx.doi.org/10.1080/15427951.2004.10129091 Zbl1098.68010
  7. [7] Perfect H., Methods of constructing certain stochastic matrices. II, Duke Math. J., 1955, 22(2), 305–311 http://dx.doi.org/10.1215/S0012-7094-55-02232-8 Zbl0068.32704
  8. [8] Saad Y., Numerical Methods for Large Eigenvalue Problems, Classics Appl. Math., 66, SIAM, Philadelphia, 2011 http://dx.doi.org/10.1137/1.9781611970739 
  9. [9] Soto R.L., Rojo O., Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl., 2006, 416(2–3), 844–856 http://dx.doi.org/10.1016/j.laa.2005.12.026 Zbl1097.15014
  10. [10] Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965 Zbl0258.65037

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