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| Paper IPM / M / 18400 |
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| Abstract: | |
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We establish tight lower bounds for the trace norm \((\|{\cdot}\|_1)\) of real symmetric and Hermitian matrices with zero diagonal entries in terms of their entrywise \(L^1\)-norms \((\|{\cdot}\|_{(1)})\). For the space of nonzero real symmetric matrices of order \(n\), we prove that the minimum possible ratio \(\frac{\|A\|_1}{\|A\|_{(1)}}\) is exactly \(\frac{2}{n}\). In the Hermitian case, this minimum ratio is given by \(\tan\left(\frac{\pi}{2n}\right)\).
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