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Matrices with eigenvectors in a given subspace
Authors:
Giorgio Ottaviani and Bernd Sturmfels
Journal:
Proc. Amer. Math. Soc. 141 (2013), 1219-1232
MSC (2010):
Primary 15A18; Secondary 13P25, 14N15, 93B25
Posted:
August 31, 2012
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Abstract: The Kalman variety of a linear subspace in a vector space consists of all endomorphisms that possess an eigenvector in that subspace. We study the defining polynomials and basic geometric invariants of the Kalman variety.
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- K. Beauchard and E. Zuazua: Large time asymptotics for partially hyperbolic systems, Arch. Rational Mech. Anal. 199 (2011) 177-227. MR 2754341
- 2.
- A. Borel and F. Hirzebruch: Characteristic classes and homogeneous spaces. I, American J. of Math. 80 (1958) 458-538. MR 0102800 (21:1586)
- 3.
- A. Compta, U. Helmke, M. Peña, and X. Puerta: Simultaneous versal deformations of endomorphisms and invariant subspaces, Linear Algebra Appl. 413 (2006) 303-318. MR 2198936 (2006m:58060)
- 4.
- W. Fulton: Intersection Theory, Springer, Berlin, 1984. MR 732620 (85k:14004)
- 5.
- W. Fulton: Young Tableaux, LMS Student Texts 35, Cambridge University Press, 1997. MR 1464693 (99f:05119)
- 6.
- M. Hautus: Controllability and observability conditions of linear autonomous systems, Indagationes Mathematicae 31 (1969) 443-448. MR 0250694 (40:3926)
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- U. Helmke and J. Trumpf: Conditioned invariant subspaces and the geometry of nilpotent matrices, in: New directions and applications in control theory, Lecture Notes in Control and Inform. Sci., 321, Springer, Berlin, 2005, 123-163. MR 2180264 (2006h:93031)
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Additional Information
Giorgio Ottaviani
Affiliation:
Department of Mathematics, University of Florence, viale Morgagni 67/A, 50134 Florence, Italy
Email:
ottavian@math.unifi.it
Bernd Sturmfels
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Email:
bernd@math.berkeley.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11404-2
PII:
S 0002-9939(2012)11404-2
Keywords:
Eigenvectors,
Kalman’s observability condition,
determinantal varieties,
Gröbner bases,
Hilbert series,
vector bundles,
Chern classes,
resolution of singularities
Received by editor(s):
December 7, 2010
Received by editor(s) in revised form:
May 23, 2011, and August 19, 2011
Posted:
August 31, 2012
Communicated by:
Harm Derksen
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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