Abstract.
A generalized Hlawka's inequality says that for any n \( (\geqq 2) \) complex numbers¶x 1 , x 2 , ..., x n ,¶¶\( \sum_{i=1}^n\Bigg|x_i - \sum_{j=1}^{n}x_j\Bigg| \leqq \sum_{i=1}^{n}|x_i| + (n - 2)\Bigg|\sum_{j=1}^{n}x_j\Bigg|. \)¶¶ We generalize this inequality to the trace norm and the trace of an n x n matrix A as¶¶\( ||A - {\rm Tr} A ||_1\ \leqq ||A||_1 + (n - 2)| {\rm Tr} A|. \)¶¶ We consider also the related inequalities for p-norms \( (1 \leqq p \leqq \infty) \) on matrices.
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Eingegangen am 23. 2. 2000
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Wada, S. Inequalities of Hlawka type for matrices. Arch. Math. 77, 415–422 (2001). https://doi.org/10.1007/PL00000512
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DOI: https://doi.org/10.1007/PL00000512