A Brauer’s theorem and related results

• Autorzy: Rafael Bru , Rafael Cantó , Ricardo Soto , Ana Urbano
• Języki publikacji: 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.

Słowa kluczowe

EN

Eigenvalues; Pole assignment problem; Controllability; Low rank perturbation; Deflation techniques

Kategorie tematyczne

• 93D15: Stabilization of systems by feedback
• 15A18: Eigenvalues, singular values, and eigenvectors

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 1
• Strony: 312-321

Daty

wydano

2012-02-01

online

2011-12-09

Twórcy

autor

Rafael Bru

• Universitat Politècnica de València

autor

Rafael Cantó

• Universitat Politècnica de València

autor

Ricardo Soto

• Universidad Católica del Norte

autor

Ana Urbano

• Universitat Politècnica de València

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
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• [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

Identyfikatory

DOI

10.2478/s11533-011-0113-0