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), 1573-1580

MSC (1991):
Primary 15A03, 15A63; Secondary 14C17, 15A42

DOI:
https://doi.org/10.1090/S0002-9939-01-06105-6

Published electronically:
February 2, 2001

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Abstract | References | Similar Articles | Additional Information

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.**A.R. Amir-Moéz,*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**18:105j****2.**I. R. Shafarevich,*Basic algebraic geometry*, Springer Study Edition, Springer-Verlag, Berlin-New York, 1977. Translated from the Russian by K. A. Hirsch; Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. MR**0447223****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****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:
https://doi.org/10.1090/S0002-9939-01-06105-6

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 A-5285.

Communicated by:
Lance W. Small

Article copyright:
© Copyright 2001
American Mathematical Society