On the variety of invariant subspaces of a finitedimensional linear operator
Author:
Mark A. Shayman
Journal:
Trans. Amer. Math. Soc. 274 (1982), 721747
MSC:
Primary 15A04; Secondary 14M15
MathSciNet review:
675077
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Abstract: If is a finitedimensional vector space over or and , the set of dimensional invariant subspaces is a compact subvariety of the Grassmann manifold , but it need not be a Schubert variety. We study the topology of . We reduce to the case where is nilpotent. In this case we prove that is connected but need not be a manifold. However, the subset of consisting of those subspaces with a fixed cyclic structure is a regular submanifold of .
 [1]
William
M. Boothby, An introduction to differentiable manifolds and
Riemannian geometry, Academic Press [A subsidiary of Harcourt Brace
Jovanovich, Publishers], New YorkLondon, 1975. Pure and Applied
Mathematics, No. 63. MR 0426007
(54 #13956)
 [2]
Thomas
Brylawski, The lattice of integer partitions, Discrete Math.
6 (1973), 201–219. MR 0325405
(48 #3752)
 [3]
F. R. Gantmacher, The theory of matrices, Vol. I, Chelsea, New York, 1959.
 [4]
Phillip
Griffiths and Joseph
Harris, Principles of algebraic geometry, WileyInterscience
[John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
(80b:14001)
 [5]
James
E. Humphreys, Linear algebraic groups, SpringerVerlag, New
YorkHeidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
(53 #633)
 [6]
Steven
L. Kleiman, Problem 15: rigorous foundation of Schubert’s
enumerative calculus, Mathematical developments arising from Hilbert
problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill.,
1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 445–482.
Proc. Sympos. Pure Math., Vol. XXVIII. MR 0429938
(55 #2946)
 [7]
M. A. Shayman, Varieties of invariant subspaces and the algebraic Riccati equation, Ph. D. thesis, Harvard University, 1980.
 [8]
Jan
C. Willems, Least squares stationary optimal control and the
algebraic Riccati equation, IEEE Trans. Automatic Control
AC16 (1971), 621–634. MR 0308890
(46 #8002)
 [1]
 W. M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press, New York, 1975. MR 0426007 (54:13956)
 [2]
 T. Brylawski, The lattice of integer partitions, Discrete Math. 6 (1973), 201219. MR 0325405 (48:3752)
 [3]
 F. R. Gantmacher, The theory of matrices, Vol. I, Chelsea, New York, 1959.
 [4]
 P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 507725 (80b:14001)
 [5]
 J. E. Humphreys, Linear algebraic groups, SpringerVerlag, New York, 1975. MR 0396773 (53:633)
 [6]
 S. L. Kleiman, Problem 15, rigorous foundations of Schubert's enumerative calculus, Proc. Sympos. Pure Math., vol. 28, Math. Developments Arising from Hilbert Problems, Amer. Math. Soc., Providence, R. I., 1976. MR 0429938 (55:2946)
 [7]
 M. A. Shayman, Varieties of invariant subspaces and the algebraic Riccati equation, Ph. D. thesis, Harvard University, 1980.
 [8]
 J. C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control 16 (1971), 621634. MR 0308890 (46:8002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206750772
PII:
S 00029947(1982)06750772
Keywords:
Invariant subspace,
Grassmann manifold,
Schubert variety
Article copyright:
© Copyright 1982
American Mathematical Society
