Proof of Wang's conjecture on subspaces of an inner product space
Authors:
Dragomir Z. Dokovic and Jason Sanmiya
Journal:
Proc. Amer. Math. Soc. 129 (2001), 15731580
MSC (1991):
Primary 15A03, 15A63; Secondary 14C17, 15A42
Published electronically:
February 2, 2001
Fulltext PDF Free Access
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Abstract: B.Y. Wang conjectured that if and are subspaces of an dimensional complex inner product space , and their dimensions are and , respectively, where , then there exists a dimensional subspace having two orthonormal bases and with and for all . We prove this conjecture and its real counterpart. The proof is in essence an application of a real version of the Bézout Theorem for the product of several projective spaces.
 1.
Ali
R. AmirMoéz, Extreme properties of eigenvalues of a
hermitian transformation and singular values of the sum and product of
linear transformations, Duke Math. J. 23 (1956),
463–476. MR 0079564
(18,105j)
 2.
I.
R. Shafarevich, Basic algebraic geometry, Springer Study
Edition, SpringerVerlag, BerlinNew York, 1977. Translated from the
Russian by K. A. Hirsch; Revised printing of Grundlehren der mathematischen
Wissenschaften, Vol. 213, 1974. MR 0447223
(56 #5538)
 3.
Bo
Ying Wang, On the extremum property of eigenvalues and the subspace
inclusion problem, Adv. in Math. (Beijing) 15 (1986),
no. 4, 431–433 (Chinese). MR 878417
(88a:15036)
 4.
, A conjecture on orthonormal bases, Linear and Multilinear Algebra 28 (1990), 193.
 5.
, A conjecture on orthonormal bases, Private communication (handwritten notes, 6 pp.), 1998.
 1.
 A.R. AmirMoéz, Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations, Duke Math. J. 23 (1956), 463476. MR 18:105j
 2.
 I. Shafarevich, Basic Algebraic Geometry, SpringerVerlag, Berlin, New York, 1977. MR 56:5538
 3.
 B.Y. Wang, On the extremum property of eigenvalues and the subspace inclusion problem, Adv. in Math. (Beijing) 15 (4) (1986), 431433 (Chinese). MR 88a:15036
 4.
 , A conjecture on orthonormal bases, Linear and Multilinear Algebra 28 (1990), 193.
 5.
 , A conjecture on orthonormal bases, Private communication (handwritten notes, 6 pp.), 1998.
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Additional Information
Dragomir Z. Dokovic
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
djokovic@uwaterloo.ca
Jason Sanmiya
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
jssanmiy@uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S0002993901061056
PII:
S 00029939(01)061056
Keywords:
Sequences of subspaces,
orthonormal bases,
B\'{e}zout theorem,
orientation of a graph
Received by editor(s):
July 30, 1999
Published electronically:
February 2, 2001
Additional Notes:
Supported in part by the NSERC Grant A5285.
Communicated by:
Lance W. Small
Article copyright:
© Copyright 2001
American Mathematical Society
