Proof of Wang's conjecture on subspaces of an inner product space (Q2706566)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Proof of Wang's conjecture on subspaces of an inner product space |
scientific article |
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20 March 2001
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sequences of subspaces
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orthonormal bases
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Bézout theorem
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orientation of a graph
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complex inner product space
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Proof of Wang's conjecture on subspaces of an inner product space (English)
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Let \(V\) be an \(n\)-dimensional complex inner product space. Let \(R_1,\ldots,R_k\) and \(S_1,\ldots,S_k\) be subspaces of \(V\) with \(\dim R_t=i_t,\) \(\dim S_t=n-i_t+1\) for \(t=1,\ldots,k,\) where \(1\leq i_1<\cdots<i_k\leq n.\) The authors show that there is a \(k\)-dimensional subspace \(W\) of \(V\) having two orthonormal bases \(\{x_1,\ldots,x_k\}\) and \(\{y_1,\ldots,y_k\}\) with \(x_t\in R_t\) and \(y_t\in S_t\) for all \(t.\) This theorem was conjectured by \textit{B. Y. Wang} [A conjecture on orthonormal bases, Linear and Multilinear Algebra 28, 193 (1990)]. The authors also prove a real version of the above theorem.
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