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Archiv der Mathematik

, Volume 77, Issue 5, pp 415–422 | Cite as

Inequalities of Hlawka type for matrices

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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.

Keywords

Complex Number Trace Norm Related Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • S. Wada
    • 1
  1. 1.Department of Information and Computer Engineering, Kisarazu National College of Technology, 2-11-1 Kiyomidai-Higashi, Kisarazu, Chiba, 292-0041 JapanJP

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