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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
