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2012 | 10 | 1 | 312-321
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A Brauer’s theorem and related results

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EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
1
Strony
312-321
Opis fizyczny
Daty
wydano
2012-02-01
online
2011-12-09
Twórcy
autor
autor
autor
Bibliografia
  • [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
  • [2] Crouch P.E., Introduction to Mathematical Systems Theory, Mathematik-Arbeitspapiere, Bremen, 1988
  • [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] Hautus M.L.J., Controllability and observability condition of linear autonomous systems, Nederl. Akad. Wetensch. Indag. Math., 1969, 72, 443–448
  • [5] Kailath T., Linear Systems, Prentice Hall Inform. System Sci. Ser., Prentice Hall, Englewood Cliffs, 1980
  • [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
  • [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
  • [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] 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
  • [10] Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-011-0113-0
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