A Brauer’s theorem and related results. (English) Zbl 1248.15007
The authors provide applications of a theorem by A. Brauer [Duke Math. J. 19, 75–91 (1952; Zbl 0046.01202)] relating the eigenvalues of an arbitrary matrix and the updated matrix by a rank-one additive perturbation. Namely, the theorem is applied to the stabilization of control systems, to the Jordan form of a matrix, and to Wielandt’s and Hotelling’s deflations.
Reviewer: C. M. da Fonseca (Coimbra)
MSC:
15A18 | Eigenvalues, singular values, and eigenvectors |
93D15 | Stabilization of systems by feedback |
93B05 | Controllability |
93B55 | Pole and zero placement problems |
15A21 | Canonical forms, reductions, classification |
Keywords:
eigenvalues; pole assignment problem; controllability; low rank perturbation; deflation techniques; stabilization; Jordan form; Wielandt’s and Hotelling’s deflationsCitations:
Zbl 0046.01202References:
[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; · Zbl 0046.01202 |
[2] | Crouch P.E., Introduction to Mathematical Systems Theory, Mathematik-Arbeitspapiere, Bremen, 1988; · Zbl 0634.03054 |
[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; · Zbl 0643.93001 |
[4] | Hautus M.L.J., Controllability and observability condition of linear autonomous systems, Nederl. Akad. Wetensch. Indag. Math., 1969, 72, 443-448; · Zbl 0188.46801 |
[5] | Kailath T., Linear Systems, Prentice Hall Inform. System Sci. Ser., Prentice Hall, Englewood Cliffs, 1980; · Zbl 0454.93001 |
[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; · Zbl 1098.68010 |
[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; · Zbl 0068.32704 |
[8] | Saad Y., Numerical Methods for Large Eigenvalue Problems, Classics Appl. Math., 66, SIAM, Philadelphia, 2011 http://dx.doi.org/10.1137/1.9781611970739; · Zbl 1242.65068 |
[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; · Zbl 1097.15014 |
[10] | Wilkinson J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965; · Zbl 0258.65037 |
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