Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


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
Published electronically: August 31, 2012
MathSciNet review: 3008870
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. 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)
  • 7. 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)
  • 8. D. Grayson and M. Stillman: Macaulay 2, a software system for research in algebraic geometry, available at
  • 9. T. Kailath: Linear Systems, Prentice-Hall, Englewood Cliffs, 1980. MR 569473 (82a:93001)
  • 10. R.E. Kalman: Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana (2) 5 (1960) 102-119. MR 0127472 (23:B518)
  • 11. C. Koutschan: Advanced Applications of the Holonomic Systems Approach, PhD Thesis, RISC, Johannes Kepler University, Linz, Austria, 2009.
  • 12. E. Miller and B. Sturmfels: Combinatorial Commutative Algebra, Springer, New York, 2005. MR 2110098 (2006d:13001)
  • 13. N.I. Osetinskiĭ, O.O. Vasil$ '$ev, and F.S. Vainshteĭn: Geometric combinatorics of Kalman algebras, Differential Equations 42 (2006), no. 11, 1604-1611. MR 2347083 (2008i:93048)
  • 14. T. Saaty: The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, McGraw-Hill, New York, 1980. MR 773297 (86f:90009)
  • 15. T. Saaty and G. Hu: Ranking by eigenvector versus other methods in the analytic hierarchy process, Appl. Math. Lett. 11 (1998), no. 4, 121-125. MR 1631150 (99b:90012)
  • 16. S. Sam: Equations and syzygies of some Kalman varieties, Proc. Amer. Math. Soc. 140 (2012), 4153-4166.
  • 17. D. Shemesh: Common eigenvectors of two matrices, Linear Algebra Appl. 62 (1984) 11-18. MR 761057 (85i:15016)
  • 18. J. Stembridge: A Maple package for symmetric functions, J. Symbolic Comput. 20 (1995), no. 5-6, 755-768. MR 1395426 (97j:05061)
  • 19. N. Tran: Pairwise ranking: choice of method can produce arbitrarily different rank order, arXiv:1103.1110.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 15A18, 13P25, 14N15, 93B25

Retrieve articles in all journals with MSC (2010): 15A18, 13P25, 14N15, 93B25

Additional Information

Giorgio Ottaviani
Affiliation: Department of Mathematics, University of Florence, viale Morgagni 67/A, 50134 Florence, Italy

Bernd Sturmfels
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720

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
Published electronically: 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.

American Mathematical Society