Archiv der Mathematik

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

Inequalities of Hlawka type for matrices



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.


Complex Number Trace Norm Related Inequality 
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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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