On some variational problems in the theory of unitarily invariant norms and Hadamard products (Q5932209)
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scientific article; zbMATH DE number 1595365
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On some variational problems in the theory of unitarily invariant norms and Hadamard products |
scientific article; zbMATH DE number 1595365 |
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On some variational problems in the theory of unitarily invariant norms and Hadamard products (English)
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12 February 2002
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matrix inequalities
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invariant norms
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Frobenius norm
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permutation matrix
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Hadamard products
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inequalities involving eigenvalues
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0.9272508
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0.9263504
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0.9102682
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0.9038122
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0.89865553
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0.8937335
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0.89128435
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0.8907242
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Let \(\|\cdot\|\) be a unitarily invariant norm, and suppose there exists a constant \(c\geq 1\) such that NEWLINE\[NEWLINEc\cdot\min \{\|W\circ G\|\mid W \text{ unitary}\} \geq \min \{\|P\circ G\|\mid P \text{ permutation}\}NEWLINE\]NEWLINE holds for every matrix \(G\) [\textit{R.-C. Li}'s conjecture in Linear Algebra Appl. 278, No. 1-3, 317-326 (1998; Zbl 0933.15032)]. Then \(\|\cdot\|\) is equivalent to the Frobenius norm \(\|\cdot\|_F\) uniformly with respect to dimension: \(c^{-1}\|A\|_F\leq \|A\|\leq c\|A\|\) for every matrix~\(A\). A similar result is obtained under a slightly different assumption.
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