Matrices with eigenvectors in a given subspace

Ottaviani, Giorgio; Sturmfels, Bernd

doi:10.1090/S0002-9939-2012-11404-2

Matrices with eigenvectors in a given subspace
HTML articles powered by AMS MathViewer

by Giorgio Ottaviani and Bernd Sturmfels PDF

Proc. Amer. Math. Soc. 141 (2013), 1219-1232 Request permission

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

Karine Beauchard and Enrique Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 177–227. MR 2754341, DOI 10.1007/s00205-010-0321-y
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
Albert Compta, Uwe Helmke, Marta Peña, and Xavier Puerta, Simultaneous versal deformations of endomorphisms and invariant subspaces, Linear Algebra Appl. 413 (2006), no. 2-3, 303–318. MR 2198936, DOI 10.1016/j.laa.2005.06.017
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
M. L. J. Hautus, Controllability and observability conditions of linear autonomous systems, Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. 31 (1969), 443–448. MR 0250694
Uwe Helmke and Jochen Trumpf, Conditioned invariant subspaces and the geometry of nilpotent matrices, New directions and applications in control theory, Lect. Notes Control Inf. Sci., vol. 321, Springer, Berlin, 2005, pp. 123–163. MR 2180264, DOI 10.1007/10984413_{9}
D. Grayson and M. Stillman: Macaulay 2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
Thomas Kailath, Linear systems, Prentice-Hall Information and System Sciences Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. MR 569473
R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mexicana (2) 5 (1960), 102–119. MR 127472
C. Koutschan: Advanced Applications of the Holonomic Systems Approach, PhD Thesis, RISC, Johannes Kepler University, Linz, Austria, 2009.
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
N. I. Osetinskiĭ, O. O. Vasil′ev, and F. S. Vainshteĭn, Geometric combinatorics of Kalman algebras, Differ. Uravn. 42 (2006), no. 11, 1532–1538, 1583 (Russian, with Russian summary); English transl., Differ. Equ. 42 (2006), no. 11, 1604–1611. MR 2347083, DOI 10.1134/S0012266106110103
Thomas L. Saaty, The analytic hierarchy process, McGraw-Hill International Book Co., New York, 1980. Planning, priority setting, resource allocation. MR 773297
T. L. 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, DOI 10.1016/S0893-9659(98)00068-8
S. Sam: Equations and syzygies of some Kalman varieties, Proc. Amer. Math. Soc. 140 (2012), 4153–4166.
Dan Shemesh, Common eigenvectors of two matrices, Linear Algebra Appl. 62 (1984), 11–18. MR 761057, DOI 10.1016/0024-3795(84)90085-5
John R. Stembridge, A Maple package for symmetric functions, J. Symbolic Comput. 20 (1995), no. 5-6, 755–768. Symbolic computation in combinatorics $\Delta _1$ (Ithaca, NY, 1993). MR 1395426, DOI 10.1006/jsco.1995.1077
N. Tran: Pairwise ranking: choice of method can produce arbitrarily different rank order, arXiv:1103.1110.